It is always lovely when you can form a convex polyhedron with a discrete optimization problem.  It gives you properties you can exploit, some that you have pointed out.  The application I'd like to point out is Scheduling.  Scheduling is pretty notorious for exploiting properties of convex polyhedra if the problem as an optimization problem can be formulated as a relaxed LP (of its IP counterpart).  Of course this usually means solving a linear program then rounding can be used (or in some realms is avoided, but instead primal-dual approaches are found at times).  

Here is an example of how (the rounding theorem in this paper is the crux of how):  https://www.researchgate.net/publication/4355176_Approximation_algorithms_for_scheduling_unrelated_parallel_machines

The above link is to a very important 2-approximation algorithm that showed how you can tackle heterogeneous computer scheduling (unrelated parallel machines) with linear programming, and still matches the best approximation factor to this date (some others have been proposed with the same approximation factor).  New results still draw from this paper to this date (see the cited in, there are a lot).

The notion of convexity can be applied to optimization problems and give us a better understanding of how well or just how we can approximate intractable problems.

Hope this helps :)!

Conference Paper Approximation Algorithms for Scheduling Unrelated Paralle Machines

Dear James. What I encountered - this methods the optimal placement of objects on a plane (for example, for optimal cutting of materials). In image processing - some segmentation techniques. The theory of convex sets is used in the economy (for example, the optimal allocation of resources).

It is always lovely when you can form a convex polyhedron with a discrete optimization problem.  It gives you properties you can exploit, some that you have pointed out.  The application I'd like to point out is Scheduling.  Scheduling is pretty notorious for exploiting properties of convex polyhedra if the problem as an optimization problem can be formulated as a relaxed LP (of its IP counterpart).  Of course this usually means solving a linear program then rounding can be used (or in some realms is avoided, but instead primal-dual approaches are found at times).  

Here is an example of how (the rounding theorem in this paper is the crux of how):  https://www.researchgate.net/publication/4355176_Approximation_algorithms_for_scheduling_unrelated_parallel_machines

The above link is to a very important 2-approximation algorithm that showed how you can tackle heterogeneous computer scheduling (unrelated parallel machines) with linear programming, and still matches the best approximation factor to this date (some others have been proposed with the same approximation factor).  New results still draw from this paper to this date (see the cited in, there are a lot).

The notion of convexity can be applied to optimization problems and give us a better understanding of how well or just how we can approximate intractable problems.

Hope this helps :)!

Conference Paper Approximation Algorithms for Scheduling Unrelated Paralle Machines

Dear Vyacheslav,

Many thanks for your observations.    If possible, please post links to optimal placement and image processing, especially in segmentation.    One obvious way to segment an image using the idea of convexity is to do the following:

superimpose a Voronoi diagram on a image.   The Voronoi diagram consists of convex polygons that cover an image.     The sample Voronoi diagrams were produced using Mathematica  (see, e.g., the 2D colour diagram, where convex polygon is filled with a single colour).   In segmenting an image using this idea, it will be necessary to decide what part of an image each convex polygon would cover.    Do you have any suggestions?

Dear Daniel,

Many thanks for your observations.   Do you have any examples of intractable problems that have been approximated using the notion of convexity?    One thing comes to mind about how one might solve the approximation problem.   Try J.M. Lien's approach, where we relax the convex set requirement and allow shapes to be partially convex.    There are many examples showing partial convexity in

http://cs.gmu.edu/~jmlien/research/app-cd/lien-dissertation.pdf

Dear James.

First of all, give reference works YG Stoyan and SV Yakovlev, which relate to the theory of optimal placement of objects on a plane:

http://link.springer.com/article/10.1007/BF02742066#page-1

http://link.springer.com/article/10.1007/BF01130367#page-1

http://link.springer.com/article/10.1007/s10559-014-9626-4#page-1

YG Stoyan and SV Yakovlev are students Rvacheva (founder of the theory of R-functions).

I remember two recent notions pertained to "convexity" in mathematical physics:

1) convexity theorem of Atiyah in symplectic manifold. 2) recent amplituhedron as convex polytopes in twistor-momentum space that simplifies the calculation of scattering amplitudes in particle and high energy physics.

Dear Vyacheslav,

As a followup to what you observed, I found the following report:

Y.G. Stoyan, N.I. Gil', A. Pankratov, G. Scheithauer, Packing non-convex polytopes into a parallelepiped, Technische Universitat Dresden, 2004:

http://www.math.tu-dresden.de/~scheith/ABSTRACTS/PREPRINTS/04-non-conv.pdf

Here the trick is to consider convex sub-regions in looking for local minima.

Dear Manouchehr,

I found the following paper:

W. Schmaltz, The Atiyah-Guillemin-Sternberg convexity theorem, 2010:

http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Schmaltz.pdf

The Atiyah-Guillemin-Sternberg convexity theorem is given (with a proof) in Section 8, p. 12.   Is this the theorem you mentioned?

Yes exactly, and a few similar articles where the relation between volume (measure) of convex polytopes toral varieties, angular momentum and algebraic geometry has been explored, the reason of my interest is the possible relation between probability amplitudes explained in the context of amplituhedron and the Koushnirenko theorem mentioned in an article by Atiah :ANGULAR MOMENTUM,CONVEX POLYHEDRA

AND ALGEBRAIC GEOMETRY. These diverse concepts may originate from a hidden  common background.

Dear Jim,

Your question forms one the kernels of mathematics! It may cove a very large part of mathematics. As for applications I can think of:

Economics: Optimisation

Algebraic Topology: 

[Günter_Ewald]_Combinatorial_Convexity_and_Algebraic_Geometry

[John_R_Stallings]_Lectures_on_polyhedral_topology

Jakob Jonsson, Simplicial Complexes of Graphs, Springer.

Various applications:

Convex Sets and Their Applications (Dover Books on Mathematics) Paperback – June 5, 2007  by Steven R. Lay

Dear @James, seems the application of convex sets is present in so many areas. I will mention the application of convex sets in the are of system science, dynamical systems and control. I have attached 3 papers bellow!  The formulation of convex optimization problems as Linear Matrix inequalities in Systems and Control is known for a long time and it is solved by MATLAB.It seems that the possibility of convex sets application is inexhaustible!

http://www-verimag.imag.fr/~maler/Papers/reach-infinity.pdf

http://www.sciencedirect.com/science/article/pii/0022247X87900953

http://infoscience.epfl.ch/record/111904/files/KKL-ECC07.pdf

Dear Costas,

Many thanks for your observations and pointers towards applications of convexity.

Dear Ljubomir,

Yes, convex sets are omni-present and very useful in quite a few applications such as those mentioned by @Costas and by you.   

Dear James,

Aside from the convex sets in Euclidean spaces either abstract or geometric used in applications (optimization, control theory, etc) and for the sake of abstract studies that are closed  by segment connectivity, to see that notion of being convex outside of mathematics, any matter that constantly moves, rotates and have frictions with an outside physical matter tends to become convex - as the edges or corners that pointed inward, reasons of concavity,  forced to disappear over time.

The universe is a convex set in which no dent will exist as it rotates and moves outward constantly.  We also see that things in nature that tend to be stable are either convex or star-shaped as the existence of dents make them loose stability and eventually disappear from existence or from their original position.

Dear James,

I am teaching the course of engineering optimization. One section is devoted to definition of convexity and convex sets. If the objective function and constraints are all convex , we will have a convex programming problem. These problems have the property that any local minimum must be a global minimum.

Real word convex optimizations problems are not very common in civil engineering and most of problems are non-convex. I have attached a Ph.D. thesis in topology optimization in which convex modeling is used:

Zhao, X. (2013). Convex modeling based topology optimization under load uncertainty (Doctoral dissertation, RUTGERS THE STATE UNIVERSITY OF NEW JERSEY-NEW BRUNSWICK).

Dear Behrouz,

Your engineering optimisation course covers some noteworthy topics and the report by A. Ben-Tal, Convex optimization in engineering: modeling, analysis, algorithms is interesting.

2011 was a bumper year for doctoral dissertations on convexity and convex optimisation.    Perhaps you will find the following thesis helpful for your course:

Martin Jaggi, Sparse Convex Optimization Methods for Machine Learning, Ph.D. thesis, 2011:

http://m8j.net/math/thesis.pdf

This thesis presents some interesting aspects of convex functions.    See, e.g., Section 2.3, starting on page 20, convex optimisation.

A good introduction to convexity and optimisation is given in

C. Vinzant, Real Algebraic Geometry in Convex Optimization, Ph.D thesis, 2011:

http://www-personal.umich.edu/~vinzant/thesisVinzant.pdf

Convexity in sets [ http://en.wikipedia.org/wiki/Convex_set ] silently implies the notion of closure: The set is closed for every specific linear combination of its elements with coefficients added to unity, ie

k1 x1+k2 x2 + ... + kn xn  belongs to  S,  for all xi,i=1,2,...,n of S, with k1+k2+...+kn=1

Thus we have a special case of closure, which is fundamental in Mathematics. I remind that both Algebra & Topology are based on closure.

The problem is that, like Taylor series expansion and linearization processes, there also exists a false attitude that all problems can be 'convexitable', ie we can find a convex approximation of them. This practice has led to big fallacies, especially in optimization algorithms.

I may add some references on convexity:

Jan van Tiel Convex Analysis An Introductory Text

Constantin Niculescu, LarsErik Persson Convex Functions and Their Applications

Ralph Tyrell Rockafellar Convex analysis

The first is an introductory text and the last two advanced.

Dear Costas,

The added references are very helpful.   I just now found an online copy of

C.P. Niculescu, Convex Functions and Their Applications.   A Contemporary Approach.  SPIN Springer, 2004, x + 250pp:

http://carma.newcastle.edu.au/jon/Preprints/Books/CUP/CUPold/np-convex.pdf

This book has 255 references, 10 pages, 236-246,

extensive keyword index, 247-250,

indexed major theorems, p. 250

helpful list of symbols, 1-3

5 details appendices, including Appendix A Background on Convex Sets, 197-206

The book itself is well-written.

@Demetris Christopoulos: ...there also exists a false attitude that all problems can be 'convexitable', ie we can find a convex approximation of them. This practice has led to big fallacies, especially in optimization algorithms.

If possible, please give examples of falacies found in optimization algorithms.   It would be interesting to see where attempts at convex approximation have failed.

Dear James, besides applications, convex sets are an object of unbelievable deep theoretical study. Convex compact sets in Rn build a Baire space, so it makes sense to speak about their topological majority (where a minority is a subset of first Baire category, and the majority is ist complement). An intersection of finitely many topological majorities is again a topological majority, so one can speak about the typical convex body. Following Victor Klee, most convex bodies are strictly convex, C1 and not C2. Many other typical properties followed, some of them discovered by Tudor Zamfirescu. The portrait of the typical convex body looks really weird, and is, to sum up, a strong existence theorem. It is really hard to imagine how the typical convex body is, when we try to put all typical properties together...

Many models, that are used every day as valid, have been built upon a hidden convexity issue.Take for example the Markowitz portfolio theory:

http://stephenkinsella.net/WordPress/wp-content/uploads/2008/02/mark91.pdf

And all utility theory (even Economics as a feild) have as requirement convexity.

A smal  example: in Greece we have three oil refiners. Can you define convexity constraints with such a small number? Maybe no, but all theory is based on convexity.

Dear @George and @Demetris,

Yes, many thanks for the suggestions about mathematical finance and econometrics.    See, for example,

E. Hazan, Efficient Algorithms for Online Convex Optimization and Their Applications, Ph.D. thesis, Princeton University, 2006:

http://ie.technion.ac.il/~ehazan/papers/thesis.pdf

See Chapter 3, starting on page 39, experiments with portfolio management.

Dear Mihai,

Many thanks for your observations about convex compact sets and convex bodies.

In terms of convex bodies, I just now found a remarkable Ph.D. thesis:

S. Taschuk, Some Inequalities in Convex Geometry, Ph.D. thesis, University of Alberta, 2013:

http://www.ualberta.ca/~staschuk/taschuk-phdthesis.pdf

This thesis gives an upper bound for the Banach-Mazur distance between an origin-symmetric convex body and the n-dimensional cube (See chapter 2, starting on page  13).

Hi James,

The design of prediction markets involves much convex analysis , for similar reasons to risk measures which George mentioned.  Convexity also underlies much of mechanism design / auction theory -- the "consumer surplus" function is always convex .  Notably, although I generally agree with Demetris' comment, in mechanism design convexity seems to come as a consequence rather than an assumption.  Lastly, there are two major appearances of convexity in statistics that I'm aware of (and surely there are many others): the cumulant function for exponential families (see Ch.3 of , and Fig.3.5), and relatedly, in scoring rules used to evaluate statistical forecasts.  The latter I find best summarized by McCarthy in his wonderful yet short 1956 paper, "Measures of the value of information":

The intuitive content of the convexity restriction is that it is always a good idea to look at the outcome of an experiment if it is free. For suppose that the experiment has two outcomes, A and A*, which would give one probabilities p and p* for the event in question. Let t be the probability that A is the outcome. If we decide not to look, our expectation is f(tp + (1-t)p*), while if we decide to look, our expectation is tf(p) + (1-t)f(p*).

It is hard not to encounter convexity everywhere once you start thinking about it!

Raf

Dear James, 

One of the most interesting problem on the space of convex bodies  the following: How can we construct a measure which is a geometric measure in the sense that invariant under rigid motions, the convex polytopes have measure zero and  the smooth bodies have positive measure? Since with respect to the Hausdorff measure this space is locally compact there are many possibility to give a Borel measure on it, unfortunatelly this is not so.  You can find some further information on this problem in the papers:

Gruber, P.M.: {\em Convex and Discrete Geometry} Springer-Verlag Berlin Heidelberg 2007.

Hoffmann, L. M.: Measures on the space of convex bodies. {\em Adv. Geom. } {\bf 10} (2010), 477--486.

Horváth, Ákos G.: Normally distributed probability measure on the metric space of norms. {\em Acta Mathematica Scientia} {\bf 33/5} (2013) 1231-1242.

Ákos

Dear Á. G. Horváth,

Many thanks for your observations and the list of papers on convex bodies and your own paper on the metric space of norms.

In your Acta Math. Sci. paper, I am especially interested in how you define the new system of bodies (p. 1240), starting with changing a body to a smooth body defined by the convex hull of the ball around the origin (nice idea!).

Dear James,

of course, there is the huge area of convex sets in differential geometry (the notion of `egg surfaces' for example, `Eiflaechen' in German) that one should not forget, with beautiful contributions by Brunn, Minkowski, Chern-Lashof, Blaschke, Santaló, Hadamard, Cohn-Vossen, Pogorelow, Herglotz...; many books on the (differential) geometry of convex sets - by Kurt Leichtweiss, Wilhelm Blaschke etc. - convex sets are  not in the main focus of research anymore, although there are still contributions. Sometimes, they still make it into textbooks (for older textbooks, this was standard material), for example, Spivak's treatise on Differential Geometry covers some of the classical material (I recall giving a talk on the Chern-Lashof theorem as a student based on his material).

Dear Ilka,

Perhaps you will find the introduction to lightlike convexity (Theorem 9.2) in the following paper interesting:

S. Izumiya, Total lightcone curvatures of spacelike submanifolds in Lorentz-Minkowski space, arXiv, 2014:

http://arxiv.org/pdf/1403.2853.pdf

Dear James,

Let me also mention the use of convex sets in statistical mechanics. In particular the density matrix (and its reduced variants) belong to a convex set with the limit point corresponding to the homogeneous ensemble. Within this theory one formulates the fundamental restrictions of N-representability in manybody quantum mechanics as well as in Liouville dynamics and associated  thermalization.

See e.g. my article in Lippert Macomber "Dynamics during spectroscopic Transitions" http://lib.ugent.be/catalog/rug01:000403795?i=121&q=%22OPTICAL+PROPERTIES%22

Best erkki

A first application is Hahn-Banach theorem and its generalizations (analytic and geometric forms). A second application, related to the first one, deals with Krein-Milman theorem and its finite dimensional form, Caratheodory's theorem. These last results have been used by Gustave Choquet in his theory of integral representation involving the extreme points of a compact convex set in a locally convex space. Finally, another application of the extreme points of a convex set (with some additional properties), is the maximum principle for convex functions: a continuous convex function on such a convex set attains its maximum at an extreme point of the set. Applications in optimization theory, moment problems and other domains have been pointed out by many authors: R.B. Holmes, "Geometric Functional Analysis and its Applications" Springer, 1975, W. Rudin, "Real and complex analysis", McGraw-Hill, Inc., 1966, R.R. Phelps, "Lectures on Choquet's theorem", D. Van Nostrand Company, Inc., Princeton, 1966, recently Mihai Putinar and many others.   

Dear Erkki,

Definitely, the use of convex sets in statistical mechanics is important, particularly in terms of what you pointed out about the formulation of the fundamental restrictions of N-representability in many body quantum mechanics.    See, for example, defining and energy cost on the convex hull of a set of particles in

F. Bavaud, Statistical mechanics of convex bodies, J of Statistical Physics 57(5), 1989, 1059-1068:

http://www.researchgate.net/publication/227230552_Statistical_mechanics_of_convex_bodies

Perhaps, you will find the following article interesting:

R.B. Israel, R.R. Phelphs, Some convexity questions arising in statistical mechanics, Math. Scand. 54, 1984, 133-156:

http://www.mscand.dk/article/viewFile/12048/10064

Perhaps others who follow this thread can suggest other applications of convexity in statistical mechanics.

Article Statistical mechanics of convex bodies

Thanks James for the references enclosed in your answer. Although they display advanced math portraits in terms of definitions beyond my actual working practice I will study them carefully.

In passing it is interesting to note that simplexes have also been used in connection with elusive Wigner solids, i.e. in associating ordered configurations of fermions in higher dimensional spaces as well as interpreting their lower dimensional projections of the fractional quantum Hall effect.

Dear @Errki and @James, wonderful book about the issue which was raised! Spectral Theory and Quantum Mechanics: With an Introduction to the Algebraic Formulation! Wonderful contents!

http://books.google.rs/books?id=cbJGAAAAQBAJ&printsec=frontcover#v=onepage&q&f=false

Dear Octav,

I just now found a copy of

Robert R. Phelps, Lectures on Choquet's Theorem, 2nd Ed., Lecture Notes in Mathematics 1757, Springer, 1966.

Strict convexity starts appearing in chapter (section) 3, Choquet's Theorem, p. 15 (Phelps has a chapter on infinite dimensional convexity in W.B. Johnson and J. Lindenstrauss, Eds., North Holland Handbook on the Geometry of Banach Spaces).

Many thanks for pointing to this book.

Dear All,

Since I have often mentioned the work of my scientific father Per-Olov Löwdin, I should also recommend his book "Linear Algebra for Quantum Theory" (1998) Wiley.

POL's teachings and the organization of about 70 summer and winter schools in Quantum Chemistry, Solid State Physics and Quantum Biology in Florida and Sweden was finally adapted (with respect the basic mathematical introduction) to the Wiley book mentioned above, at the end of his life. The book is a pedagogical master piece and in addition there are some interesting sections on convex functions, inequalities, and convex set of system operators etc.

Best

erkki

Abstract convexity allow to create a bridge from Geometry and pure Combinatorics (e.g., Graph Theory) embracing new cross development in those, and other areas of research. See for examples

http://dml.cz/dmlcz/107984

http://www.sciencedirect.com/science/article/pii/S0195669807002181

In the paper by  J. Nesetril and R. Strausz,

Universality of separoids, Archivum Mathematicum 42(1), 2006, 85-101

with the URL helpfully given by @Ricardo Strausz.    The following example (p. 86) is given:

In Example 2,  J. Nesetril and R. Strausz introduce a family of convex sets F (instead of points) in a d-dimensional (normed) linear space $E^d$.  Let S be a finite set endowed with a symmetric relation define on its family of subsets (in other words, S is a separoid).   Let S(F) denote a separoid defined relative to S and F.   That is, a pair of disjoint subsets are separated, provided the disjoint subsets are not a Radon partition.   In particular, two subsets A,B of the family of convex sets F are separated, provided there exists a hyperplane so that all members of A are one side of the hyperplane and all members of B are on the other side the hyperplane.   That is, S(F) is a collection of separated subsets in F. This leads to the following reasonable theorem:

Theorem Given a separoid S in $E^d$, there exists a family of convex sets in F such that S is isomorphic to S(F).

Perhaps one of the followers of this thread will offer a proof of this Theorem. 

Another good example is Pakistan's flag. See

http://en.wikipedia.org/wiki/Flag_of_Pakistan

Dear @Ismat Beg,

Another place to look for convex sets is in a Delaunay mesh, a triangulation of a set of points called sites.    Several examples of meshes containing many convex polygons are attached to this post.

The first example shows the surface of a hand (itself not convex) covered with triangles that are convex.

The second example shows a Delaunay mesh covering a convex surface.

The third example shows the sites (selected points) used to construct the Delaunay triangulation of a square.    This example also includes a circumcircle that touches (passes through) the vertices of the inner and outer triangles along perimeter of the circle.   It would be interesting to "lift" this triangulation (from the centre) to exhibit a 3D view of the convex shapes surrounding the centre.

HI, herewith an application. Given an architectural plan, e.g,  the Alhambra plan

http://www.learn.columbia.edu/ma/htm/dj_islam/ma_dj_image_alh_plan01.htm

Is it possible to automatically partition the indoor space into many convex spaces? the set of convex spaces must contain the least number of fattest convex spaces.

In this recent paper, http://arxiv.org/abs/1502.03554

we manually identify all convex spaces. The manual process goes like this, start with the first fattest space, proceed with the second fattest space, and so on until the indoor space is fully covered by the least number of fattest spaces. However, we are seeking an automatic solution. Any advice is highly anticipated.

Cheers.

Bin

@Bin Jiang : Is it possible to automatically partition the indoor space into many convex spaces?

Yes, provided that a scale model of the indoor space.   The scale model can be either in 2D or 3D.   One way to solve the partition problem is to separate the parts of space into polygon regions.   Assuming that each polygonal region can be drawn in the plane, then a number of choices can be made to obtain the partition of each polygonal region.   Here is one of the choices:

Voronoi mesh:   Select a set S of sites (points used to construct a mesh).    Let p be a site in S.   Let $\mathbb{R}^2$ denote a 2D plane.   Let ||x - p|| denote the Euclidean distance between points p and x, The construct a Voronoi region $V_p$ defined by

\[

V_p = \left\{ x \n \mathbb{R}^2: ||x - p|| \leq ||x - q||, \forall q\in S \right\}

\]

Each point in the 2D plane has a least one nearest point in S.   Hence, x lies in at least one Voronoi region $V_p$.    This means that the voronoi regions cover the entire plane.   And each Voronoi region is a convex polygon.    A sample Voronoi mesh is shown in the attached image.

Hi, the solution sounds excellent if the indoor space is fully empty. However, the indoor space has many barriers such as walls and pillars; see the example of the Alhambra: http://www.learn.columbia.edu/ma/htm/dj_islam/ma_dj_image_alh_plan01.htm

By the way, how to insert a picture into this editor tool? copy and paste seems does not work. Thanks.

Bin

An interesting application of convexity is on Exponential families by Ole Barndorff-Nielsen:

Ole Barndorff-Nielsen

http://en.wikipedia.org/wiki/Ole_Barndorff-Nielsen

Exponential family

http://en.wikipedia.org/wiki/Exponential_family 

@Costas Drossos: An interesting application of convexity is on Exponential families by Ole Barndorff-Nielsen.

Many thanks for pointing to the work by Ole Barndorff-Nielsen, who introduces a natural (vector) parameter $\nu$ in a probability density function that models a single-parameter family that is a set of probability distributions.   It is mentioned that the set of values of $\nu$ for which the function $f_X(x;\theta)$ is finite is called the natural paramter space.   And that it can be shown that the natural parameter space is always convex.   Do you know of a proof of this assertion?

I played with the idea of functions with vector parameters independent of PDFs.   I found the following Mathematica 10 script that may be of interest to followers of this thread:

Plot3D[Sqrt[1 - x^2 - y^2], {x, -1, 1}, {y, -1, 1},

MeshShading -> {{Yellow, Green}, {Pink, Red}}, Mesh -> 8]

Each cell in the mesh produced by this script is a 3D convex quadrilateral (see the attached image).

@Bin Jiang: Hi, the solution sounds excellent if the indoor space is fully empty. However, the indoor space has many barriers such as walls and pillars….

Many thanks for pointing this out.    There is a solution to the problem of partitioning an indoor space (with barriers such as walls and pillars) with convex spaces.    The basic approach hinges on judiciously sectioning an indoor space with barriers so that each barrier is a section boundary.    For simplicity, assume that a barrier such as a wall has the thickness of a line segment in geometry.   The a wall provides an ideal boundary of a section of an indoor space.    Then a Voronoi mesh can be superimposed on each wall-bounded section, i.e., each wall-bounded space is tessellated with a Voronoi mesh, provided a set of special points called sites is selected.    A common choice of sites is the set of centroids of the segments of a section of indoor space. 

In the case of pillars in an indoor space, assume that the diameter of each pillar has the diameter of a point in geometry, i.e., a zero-length diameter.    Then use the pillars as a source of corner sites (instead of centroid sites).

@Bin Jiang: By the way, how to insert a picture into this editor tool?   To insert a picture,  do the following:

1.   click on the file button in the menu bar at the bottom of a message.

2.   select the picture file in one of your local directories.

3.   let the editor upload your picture after you have selected a picture file name by highlighting it with your cursor.

Then the uploaded picture will appear at the bottom of a message.

The suggestion sounds very promising, so thanks a lot...

The following picture shows convex spaces we manually identified for part of the Alhambra plan: barriers in gray, convex spaces in white, and their centroids in red. The dot sizes show the degrees of life for individual convex spaces as developed in this paper: http://arxiv.org/abs/1502.03554

This is also to test how to insert a picture into the editor tool.

@Bin Jiang: The following picture shows convex spaces we manually identified for part of the Alhambra plan: barriers in gray ...

I just now experimented with your floorplan image and did the following things:

1.  found the corners in each part of the floorplan.

2.  constructed Voronoi mesh, using the corners as generating points (sites).   The resulting Voronoi mesh consists of different polygons.

3.  superimposed the Voronoi mesh on the floorplan.

4,  constructed a Delaunay mesh, using the corners again as generating points.  The resulting Delaunay mesh consists entirely of triangles with varying area, depending the distance between pairs of generating points.

5.  superimposed the Delaunay mesh on the floorplan.

See the attached image for the result.   You may want to consider separate meshes not superimposed on each other.

Another interesting application of convex sets is in the approximation theory. If the approximation set or the norm are strictly convex, then the best approximation is unique.See the second chapter of the book "Approximation theory and methods" by M.J.D.POWELRegards.

@Ahmad Shirzadi: Another interesting application of convex sets is in the approximation theory.

Excellent!

As a followup in terms of convexity in solving computer vision problems, consider

C. Olsson, Global optimization in computer vision: Convexity, cuts and approximation algorithms, Lund University, 2009:

http://www2.maths.lth.se/vision/publdb/reports/pdf/olsson-phd-09.pdf

The basic idea is to define a transformation so that the mapping of the measurement points comes as close as possible to their corresponding convex sets.

See Corollary 7.3.3, page 150:  The set of all of all subgradients of an $\theta$ at $\sigma_0$ is a convex set, which is also a superset of a convex hull.   A geometric interpretation of sub gradients is given in Fig. 7.1, page 149.   There are 342 instances where convexity is considered in this thesis.

Dear James.

I found the following applications of convexity: 

- convex optimization in multi-criteria problems.

- optimizing the energy flow in the smart grids to solve the problem of balance and stability of the electrical networks.

- Surgical operations to cut the tumor tissues in the brain.

-Tumors in the treatment area by microwave

- in the field of robotics for searching optimal path to avoid cavities that reflect the non-convexity of the obstacle.

Dear Omar,

The applications that you have listed are excellent.    Please consider giving a URL for each application so that we can follow up on what you have found.

a book on convex sets applied to dynamic systems and control theory:

Skelton, Iwasaki, Grigoriadis "A Unified Algebraic Approach to Control Design"

Taylor and Francis, 1998

@Robert E Skelton: a book on convex sets applied to dynamic systems and control theory...

This is quite a surprising and very promising path to take in terms of applications of convexity and convex sets.

Another example to consider is the reduction of system and control theory to a handful of convex and quasi convex optimization problems,  introduced in

S. Boyd, L.E. Ghaovi, E. Feron, V. Aladrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM (Society for Industrial and Applied Mathematics), Philadelphia, 1994:

https://web.stanford.edu/~boyd/lmibook/lmibook.pdf

There are a 161 instances where convexity is considered in this book.    See Section 1.1, starting on page   1, for an overview of convex and quasi convex optimization problems involving linear matrix inequalities (LMIs).    As it turns out, the LMIs in system and control theory can be formulated as convex optimization problems that are amenable to computer solution.

Linear matrix inequalities or LMIs are the focus of Chapter 2, starting on page 7.    See Section 2.2.4, starting on page 11 for a convex problem.

IN OPERATIONS RESEARCH WE USE THE CONCEPT "CONVEX SET".

THE FEASIBLE REGION OF ANY LINEAR PROGRAMMING PROBLEM IS A CONVEX SET AND AS AN APPLICATION WE PROVE SEVERAL RESULTS AND THESE RESULTS WILL BE APPLIED IN SIMPLEX METHOD, AS WELL AS IN REVISED SIMPLEX METHOD

I do not remember if we cite inequalities. Anyway convexity has nice applications in proving various inequalities. For example see:

• Jensen's inequality

http://en.wikipedia.org/wiki/Jensen%27s_inequality

•  Chung-Lie Wang Convexity and Inequalities, http://www.sciencedirect.com/science/article/pii/0022247X79902944

Dear James.

I found the following links for the applications of convexity I proposed above: 

For convex optimization in multi-criteria problems, there is some interesting links

https://people.kth.se/~andersf/doc/paretofront.pdf

http://ttic.uchicago.edu/~mahdavi/publications/multiobjective_nips_2013.pdf

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.6672

For optimizing the energy flow in the smart grids to solve the problem of balance and stability of the electrical networks.see 

http://www.diva-portal.org/smash/get/diva2:602608/FULLTEXT02

http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6558479

http://www.ee.columbia.edu/~lavaei/Generalized_Net_Flow_Conf.pdf

For Convex Surgical operations to cut the tumor tissues in the brain: 

www.google.com/patents/US20090124975

http://neurosurgery.stanford.edu/patient_care/brain_tumor.html

http://brain-tumor.org/BTC_Meningioma.aspx

ftp://ftp.esat.kuleuven.be/sista/adevos/thesis_lukas/phd.pdf

http://www.robomechjournal.com/content/pdf/s40648-014-0006-7.pdf

-Tumors in the treatment area by microwave

http://www.rf-ablation.engr.wisc.edu/papers/Yang_TissueTemp_MWA.pdf

http://www.cancertreatmentwatch.org/reports/holt1.pdf

I'm looking forward to find other references.

Thanks for fine references dear @Omar. Here are some more results on application of convex sets in risk management!

http://www.hussmanfunds.com/wmc/wmc091102.htm

http://www.continuitycentral.com/feature1296.html

One application of convex sets  in structural engineering

The attached investigates the reliability assessment of structures exhibiting both stochastic and bounded uncertainties by using a probability and convex set mixed model. The safety measure of a structure is quantified by a reliability index defined by a nested minimization problem. An iterative procedure is developed for seeking the worst-case point and the most probable failure point in the standard uncertainty space. Numerical examples are given to demonstrate the applicability of the probability and convex set mixed model representation in the structural reliability assessment, as well as to illustrate the validity and effectiveness of the proposed numerical method.

http://www.sciencedirect.com/science/article/pii/S0045794909001709

Thanks for all;

Additional applications in :

Equilibrium theroy and Economics

      http://www.wiwi.uni-bonn.de/john/Publikationen/UsesGCGM.pdf

dynalic systems and interacting Gases

      http://www.math.toronto.edu/mccann/papers/short.pdf

The list is not limited, we can discover evry day many applications, so that we can say: the good life is the convex one !!!

Dear Omar, Behrouz Ahmadi-Nedushan, Robert E Skelton, Bhavanari Satyanarayana, Costas Drossos, and Ljubomir Jacić,

Many thanks for the excellent links you have given for applications of convex sets.   It is surprising that such a simple idea (convexity) can have such a profound impact and have so many diverse applications.

In addition to the applications already mentioned, there are also convex hulls (smallest convex set containing a nonempty set) and their approximation to consider.   For example, consider the gesture recognition system in

M.A. Buautista, A.H.-Vela, S. Escalera, L. Igual, O. Pujol, J. Moya, V Violant, M.T. Aguera, A gesture recognition system for detecting behavioural patterns of ADHD, arXiv 1410, 2014, no. 4485v2,:

http://arxiv.org/pdf/1410.4485.pdf

Another important application of convexity is in digital geometry (also called Compuational Geometry):

R. Klette, A. Rosenfeld, Digital Geometry.  Geometric Methods for Digital Picture Analysis, 2005:

http://www.citr.auckland.ac.nz/~rklette/Books/MK2004/References.pdf

Yet another interesting application of convexity is in the use of convex hulls in digital image analysis.   Consider, for example,

R. Rubinstein, Analysis and synthesis modelling methods in image processing, Ph.D. thesis Technion, 2012:

http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-get.cgi/2012/PHD/PHD-2012-03.pdf

See the MAP-synthesis defining polytope introduced on page 40 (see, especially, page 60).   This leads to dictionaries for sparse representation modelling.

Hi James,

An application that is widely used, is mapping the earth. Because the earth is convex, we can map that 3D object on a 2D plane, the Mercator projection for example.

Being convex is not necessary, but it surely helps. Such a 2D projection can be made whenever the polyhedron is 'star shaped', that is, if at least one point in the interior can be found, from which the whole surface can be 'seen'. A convex polyhedron has the advantage that any point in the interior can be chosen as the origin to create the map. Usually the center of mass is chosen.

I have used this many times when giving a Power Point talk featuring prostates and bladders. Those 3D organs can not be shown in 3D when giving a presentation, so I resorted to a Mercator projection of them. The same holds for scientific papers: they still want them to be readable in 2D. (I wonder when that will change...)

Cheers, Lambert

Hi Lambert,

Until you wrote, I had not thought of the earth as convex.   I am guessing that you mean the idealization of the earth without bumps and viewed as a perfect sphere.    Is this what you meant?

And what you observed about mapping the earth based on the assumption that the earth is convex, is great!

Cheers, Jim

Dear  James and followers,

The question and discussion  are very interesting. The notion of convexity is fundamental in mathematics  and you mention  important applications.  I will try  to  give a few modest examples from my experience.

We observed the following in communication with M. Albijanic. Suppose that $f$ is continuous function on closed interval $[a, b]$ and differentiable on open interval (a, b). Then f is strictly convex iff for every points y, z in [a, b] there is an unique c such that $f (z) -f (y) = f' (c) (z-y)$. This statement has visual geometric interpretation. Perhaps, anyone knows the right reference for this statement. For elementary properties of convex functions see for example: Miodrag Mateljevic, Marek Svetlik, Miloljub Albijanic, Nebojsa Savic, generalizations of the Lagrange mean value theorem and applications, Filomat 27: 2 (2013).  In this paper we also outline applications   in economy.

We can give a similar characterization of strictly convex surfaces using curvature and the Gauss map.

Really, a twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. Visually, a twice differentiable convex function "curves up", without any bends the other way (inflection points). If its second derivative is positive at all points then the function is strictly convex, but the converse does not hold.

For example, it seems that we can prove

Theorem 1. Let $z=f(x,y)$ be a real-valued function defined on a

domain $D$ in $\mathbb{R}^2$. Then it defines a surface $W$ in $\mathbb{R}^3$. If its

Gauss map $N$ is injective, then

$W$ is convex.

Tere is a version of this result for Equipotential surfaces in electrostatics and fluid mechanics.

$N(p)$ is a unit vector orthogonal to $W$ at $p$, namely the normal vector to $W$

at $p$.

If $p=(p_1,p_2,p_3)$, $X(p)= (f'_x, f'_y,1)$ and $N(p)=X(p)/|X(p)|$.

Since we compute $f'_x, f'_y$ at ($p_1,p_2)$, here we can consider

that Gauss map is defined on $D$.

Theorem 2. If $z= f(x_1,x_2,\cdots,x_n)$ defined a hyper surface

$W$ in $\mathbb{R}^{n+1}$ and Gauss map is injective, then $W$ is convex.

Do you think that we can apply this?

The Gauss map (named after Carl F. Gauss) maps a surface in Euclidean

space $\mathbb{R}^3$ to the unit sphere $\mathbb{S}^2$. Namely, given a surface $X$ lying in $\mathbb{R}^3$, the

Gauss map is a continuous map $N: X \rightarrow \rightarrow \mathbb{S}^2$ such that $N(p)$ is a unit vector

orthogonal to $X$ at $p$, namely the normal vector to $X$ at $p$.

The Gauss map can be defined (globally) if and only if the surface is

orientable, in which case its degree is half the Euler characteristic.

best,

Miodrag

@Miodrag Mateljević: Theorem 2. If $z= f(x_1,x_2,\cdots,x_n)$ defined a hyper surface

$W$ in $\mathbb{R}^{n+1}$ and Gauss map is injective, then $W$ is convex.

A beautiful result!   In fact, both results in your post are both beautiful and important.

Yes, it should be possible to apply your result (in Theorem 2) in a number of different ways.   For example, if $W$ is defined in a finite topological space equipped with a proximity, it would be possible to detect hyper surfaces that are either close to or remote (far from) each other.   In effect, this approach would lead to (1) clusters of proximal hypersurfaces, (2) classification of hypersurfaces in terms of the clusters they belong to.

Dear James and followers,

I will  add  another example.    In the paper

THE TEACHING OF MATHEMATICS

2011, Vol. XIII, 1, pp. 15–29

A PROBLEM FROM THE PISA ASSESSMENT

RELEVANT TO CALCULUS by

Miodrag Mateljevic, Aleksandra Rosic and Marek Svetlik,

which is continuation of our previous papers    (see M. Mateljevic, M. Svetlik, A contribution to the development of functional thinking related

to convexity, The Teaching of Mathematics, XIII, 1 (2010), 1–16.), we discuss one of

the PISA problems where students are assumed to understand the given functional

dependence as well as to select the intervals of convexity and concavity of the

drawn graph. In particular, we give a geometric lively characterization of convexity.

I will add another example related to geometry of planar curves and a mechanical interpretation.

Theorem1.  If for a planar curve Gauss map   is injective, them  the curve is stricly convex.

We have a  mechanical interpretation of Theorem 1.

If v=v(t) is velocity at a moment t, where t is time, then unit velocity vector is defined as V*(t)= v(t)/ |v(t)|.

Theorem2.    If an object  (a point) makes planar motion  and if  the unit velocity  vector as function

of time is injective on  some time interval I,  then  the  trajectory is strictly convex curve on I..

To James  Peters,

Thank s for suggestion:

''In effect, this approach would lead to (1) clusters of proximal hypersurfaces, (2) classification of hypersurfaces in terms of the clusters they belong to.''

 Do you know some references ?

To Miodrag Mateljević,

The idea of clusters of proximal hypersurfaces has its roots in a number of my recent papers and my book on the Topology of Digital Images, Springer, 2014.

Think of a hyper surface $f(x_1,x_2,\cdots,x_n)$ as a set of points.   Let $f$ and $g$ be a pair of hyper surfaces on finite topological space X endowed with a proximity $\delta$.  Assume $\delta$ is a Wallman proximity.  Then $f \delta g$ ($f$ is near $g$), provided 

\[

\mbox{cl} f \cap \mbox{cl} g \neq \emptyset.

\]

Continuing this in this manner, using hyper surface $f$ as a  template or hub in a cluster of hyper surfaces, then find all hyper surfaces $h$ such that $f \delta h$.  This forms a cluster of proximal hyper surfaces.  

For another reference, see my paper (co-authored with C. Guadagni) on strong proximity on smooth manifolds.

Dear James, this is an interesting question!

Are you interested in symplectic geometry: the Atiyah-Guillemin-Sternberg convexity theorem (the convexity of the image of the moment map)?

http://www.math.cornell.edu/~apires/convexitytalk.pdf

@Mohamed amine Bahayou: Are you interested in symplectic geometry…?

Yes, definitely, I am very interested in symplectic geometry.    Recently, I have been working with manifolds with boundaries, manifolds with corners,  pullback maps (p. 59).   See, for example, 

J. Watts, Diffeologies, Differential Spaces, and Symplectic Geometry, Ph.D. thesis, University of Toronto, 2012:

http://www.math.toronto.edu/jwatts/storage/ut-thesis.pdf

In terms of the  Atiyah-Guillemin-Sternberg convexity theorem (very interesting!), see

A.R.P. Pires, Origami manifolds, MIT, Ph.D. thesis, 2010:

http://dspace.mit.edu/bitstream/handle/1721.1/60200/681966912-MIT.pdf?sequence=2

Start on page 11, last paragraph about the moment image being convex and, especially, Section 3.1, starting on page 31.

I will give a few inequalities related to convexity.

1. Karamata's inequality: Let I be an interval of the real line and let f denote a

real-valued, convex function defined on I. If x1, . . . , xn and y1, .

. . , yn are numbers in I such that (x1, . . . , xn) majorizes (y1, . .

. , yn), then

f(x_1)+\cdots+f(x_n) \ge f(y_1)+\cdots+f(y_n).

2. Petrovic's inequality: Let f : [0,\infty) \rightarrow R be a convex function, and

(x_i)_{i=1}^n, be a sequence of positive numbers. Then the

inequality f(x_1) + ... + f(x_n) \leq f(x_1 + ... + x_n) + (n -

1)f(0) holds.

3. A function f : (a, b) \rightarrow R is convex (strictly

convex) if and only if its divided difference \Delta_f (x, y) is

increasing (strictly increasing) in both variables. An analogous

assertion is valid for concave (strictly concave) functions.

As an  application of Karamata's inequality we can prove.

4. Prove that for arbitrary positive numbers a, b and c the

inequality \frac{1}{a + b} + \frac{1}{c + b}+ \frac{1}{c + a}\leq

\frac{1}{2a} + \frac{1}{2b} + \frac{1}{2c} holds.

5.For x,y,z \geq 0,

x^3 + y^3 + z^3 + 3xyz \geq x^2y + xy^2 +

y^2z + yz^2 + z^2x + zx^2.

Helly's theorem is a nice theorem related to convex sets.

Helly's theorem is a basic result in discrete geometry describing the ways that convex sets may intersect each other.

Let X1, ..., Xn be a finite collection of convex subsets of $R^d$, with $n > d$. If the intersection of every $d+1$ of these sets is nonempty, then the whole collection has a nonempty intersection; that is,

$\bigcap_{j=1}^n X_j\ne\varnothing$.

For example in planar case $d=2$, if we have four convex subsets such that the intersection of every $3$ of these sets is nonempty, then these sets have common point.

For infinite collections one has to assume compactness.

The convexity concept is applied on mathematical programming for obtaining the optimal solution , for example , a local solution  of a convex function is global solution and a global solution is unique if a function is strictly convex 

The following results  named as  Lucas Teorem (related to convex hull of zeros of a complex polynomial)    has interesting applications:

Lucas Teorem.

Let $P$ be a complex polynomial. The smallest convex polygon

that contains the zeros of $P$ also contains the zeros of $P'$.

In particular, if zeros of $P$ are in the unit disc, then also

zeros of $P'$ are in the unit disc.

Let $P$ be a polynomial of dgree $n$ and $a_1,\cdots,a_n$ be zeros of $P$. Then

\begin{equation}

\frac{P'(z)}{P(z)}=\frac{1}{z-a_1}+\dots +\frac{1}{z-a_n}

\end{equation}

and therefore

\begin{equation}

\frac{\overline{P'(z)}}{\overline{P(z)}}=\frac{z-a_1}{|z-a_1|^2}+\dots

+\frac{z-a_n}{|z-a_n|^2}\,.

\end{equation}

Let $P'(a)=0$.

Set $\lambda_k= \frac{1}{|a-a_k|^2} $ and $A=\sum \lambda_k = \frac{1}{|a-a_1|^2}+\dots

+\frac{1}{|a-a_n|^2}$. Then

$$a = A^{-1}\frac{a_1}{|a-a_1|^2}+\dots +\frac{a_n}{|a-a_n|^2}\,.$$

Hence $a$ belongs to convex hull of $a_1,\dots,a_n$.

@Abd El-monem Abd dEl-hamee Megahed: a local solution of a convex function is global solution and a global solution is unique if a function is strictly convex.

Your observation packs a whollop, i.e., says a lot and to a very important aspect of convexity.   To bring others into this, permit me to add E.W. Weisstein's definition of a convex function, which is a very interest "animal": A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval. from

http://mathworld.wolfram.com/ConvexFunction.html

Sixteen examples of convex functions are given in

https://en.wikipedia.org/wiki/Convex_function

Perhaps the followers of this thread can suggest other examples of convex functions.

Let K be a compact subset in  R^n and f a real continuous function on K. Let C be the convex hull of the graph of f and p(x):=inf {t; (x,t) is in C}, q(x)=sup{t; (x,t) is in C}. Then p is convex on the the convex hull co(K) of K, q is concave on the same subset and min{p(x); x is in co(K)} = min{f(y); y is in K), max{q(x); x is in co(K)} = max{f(y); y is in K}. Proceeding in this fashion, one reduced optimization problems for continuous functions defined on (non convex) compact subsets to convex minimization (respectively concave maximization) problems. It is very probable that such methods work in a more general setting. 

a convex function is very important  for obtaining the optimal solution of mathematical programming

A New Type of Application of Convex Polytope Theory in Economics 

Vyacheslav Lyashenko 1 (the number after the name indicates the post number in this question box) and Costas Drossos 10 mention applications of theory of convex sets in economy. They are thinking of optimization problems. Demetris Christopoulos 22 and George Stoica 23 mention financial engineering (or mathematical finance), Rafael Frongillo 26 the design of market predictions, Omar Bouattane 61 the equilibrium theory in economics. 

However, there is one (perhaps more important and interesting) application of the theory of convex sets in economics. It is the international trade theory. Ricardian trade theory (including Ricardo-Sraffa trade theory) is in fact a theory of special type of convex polytopes: world production possibility set and it maximal frontier. In particular, the theory of Ricardian trade economy (which excludes trade of input goods) has a beautiful theory as an application of subtropical convex sets. Thus it is a theory combined of classical convex polytopes and tropical convex polytopes. The theory may also give a good example for the theory of convex optimization, which Omar Bouattane 52 and 58 mentioned.

 See the paper below. The field is still young and I think it will develop rapidly if some specialists in convex polytope theory participate in it. 

https://www.researchgate.net/publication/280646264_International_trade_theory_and_exotic_algebra

Article International trade theory and exotic algebra

Dear Miodrag Mateljević,  

I love Helly theorem very much. It is so beautiful, simple, and subtle. I was very happy when I found an application of the theorem in my trade theory. See Proposition 7.4 (p.72). 

https://www.researchgate.net/publication/236020268_Subtropical_Convex_Geometry_as_the_Ricardian_Theory_of_International_Trade_Contents  

Article Subtropical Convex Geometry as the Ricardian Theory of Inter...

@Yoshinori Shiozawa: I love Helly theorem very much. It is so beautiful, simple, and subtle.

Yes, I definitely agree with you about the beauty, simplicity and subtlety of Helly's theorem for families of convex bodies in R^d.   Helly's theorem and its application is covered in Chapter 11 in my new book:

J.F. Peters, Computational Proximity.   Excursions in the Topology of Digital Images.  Springer, 2015 [now in press].

There are also a number of important studies of Helly's theorem on RG.   See, e.g.,

https://www.researchgate.net/publication/242648091_Helly's_theorem_and_its_relatives

https://www.researchgate.net/publication/266970982_A_Fractional_Helly_Theorem_for_Boxes

https://www.researchgate.net/publication/38334457_Hellys_theorem_and_minima_of_convex_functions

Article Hellys Theorem and its relatives

Article A Fractional Helly Theorem for Boxes

Article Helly’s theorem and minima of convex functions

Dear  James Peters,

thank you for your comment and information on literature. This is the first time that I know the personal life of Edward Helly. 

As for my paper  Subtropical Convex Geometry as the Ricardian Theory of International Trade Contents, I am planning to publish it in a form of thin book, but I have to add one or two more sections.

A short version (that I referred in the last post of the previous page) of the above paper was published in Evolutionary and  Institutional Economics Review 12(1): 172-212.  

I want to know if Jones theorem (the proposition 7.4 of Subtropical Convex paper and the theorem 6 of International Trade paper) is known previously. Jones himself did not gaive the proof of existence.

Yoshinori Shiozawa

There is a lot of literature on generalized convexity in metric, semi-metric and pseudo-metric spaces with applications to fixed point theory.

The definition of general convexity in such spaces usually assume:

A. Intersection of genl convex sets are genl convex sets.

B. Closed sets of some form are genl convex

But weaker forms are also useful.

Prof Peters  posted  a very interesting question.

The number of answers submitted to the question  and   the number of useful votes that those answers receive  are significant.

We   would like to  note  an    interesting phenomenon : if a function f  defined  on an segment [a,b] is convex and the derivative   f'  exists  on (a,b), then    f' is continuous on (a, b).

Any convex function $f(x)$ on [a, b] is continuous on (a, b) and

has a finite right derivative $f_{+}(x)$ and a left derivative

$f_{-}(x)$ at each point $x \in (a; b)$. Moreover, for all $x

\in (a; b)$, $f_{-}'(x)\leq f'_{+}(x)$, the equality occurring and

yielding the derivative $f'(x)$ everywhere, except possibly a

countable number of points inside $(a, b)$. Wherever it exists,

$f'(x)$ is a non-decreasing function of $x$.

Proposition A.  Let $f(x)$ be a continuous function on a closed interval $[a,b]$

and differential on the open interval $(a,b)$. In addition, if

$f$ is convex or concave, then  $f'$ is

continuous . This result may exist

in some forms in the literature, but we did not find it.

Dear followers, do you know the origin of this result?

best,

MM

@Miodrag Mateljević: if $f$ is convex or concave, then $f'$ is continuous. 

A good place to look for this result is

Steven Taschuk, Some Inequalities in Convex Geometry, Ph.D. thesis, 2013:

https://www.ualberta.ca/~staschuk/taschuk-phdthesis.pdf

See Section 4.3 (Continuity), starting on page 36.

Compactness of bounded functions.

The pare LAGRANGE'S THEOREM, CONVEX FUNCTIONS AND GAUSS MAP which is related to the question,  is enclosed.

As one of the main results

we prove that if $f$ has Lagrange unique

property then f is strictly convex or concave (we do not assume

continuity of the derivative), Theorem 2.1. We give two

different proofs of Theorem 2.1 (one

mainly using Lagrange theorem and the other using Darboux

theorem). In addition, we give a few characterizations of

strictly convex curves, in Theorem 3.1.

As an

application of it, we give

characterization of strictly convex planar curves, which have

only tangents at every point,

by injective of the Gauss map. Also without the

differentiability hypothesis we get the characterization of

strictly convex or concave functions by two points property,

Theorem 4.1.

By the convexity we can prove that the local minimum is global minimum.

and if the function is strictly convex , then the global minimum is unique.

It is interesting that   there is 

Kachurovskii's theorem,   whicih   is a theorem relating the convexity of a function on a Banach space to the monotonicity of

its Fréchet derivative.

Statement of the theorem:

Let K be a convex subset of a Banach space V and let f maps  K into  the  extended real-line, that is Fréchet differentiable  at each point x in K. (In fact, df(x) is an element of    the continuous dual space V*.) Then the following are equivalent:

f is a convex function;

(i)    for all x and y in K,

\mathrm{d} f(x) (y - x) \leq f(y) - f(x);

(ii)    df is an (increasing) monotone operator, i.e., for all x and y in K,

\big( \mathrm{d} f(x) - \mathrm{d} f(y) \big) (x - y) \geq 0.

If f is a function of real variable on some interval that is differntiable then (i) f'(x) (y-x)\leq f(y)-f(x),

(ii) f'(x)-f'(y) (y-x)\geq 0.

Thus (ii) means that f' is not decreasing.

Do you know some applications of  Kachurovskii's theorem?

References

Kachurovskii, I. R. (1960). "On monotone operators and convex functionals". Uspekhi Mat. Nauk 15 (4): 213–215

Showalter, Ralph E. (1997). Monotone operators in Banach space and nonlinear partial differential equations.

Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. 80. ISBN 0-8218-0500-2.

MR 1422252 (Proposition 7.4)

Dear George,

Many thanks for the very nice result.

Please consider putting together an example to illustrate a real function that satisfies the theorem.

Dear George  and James,

On a first glance, the  function f(x)=|x| is convex, but there is no derivative at 0.

More intriguing is  the following example:

Let $(r_k)$ be sequence of all rational numbers. Function

f(x)= \sum_1^\infty |x-r_k|/2^k is convex, but there is no derivative at rational points.

There is   the Integral representation  for convex functions.

Convex functions on the real line are expressible as integrals of one-sided derivatives.

The ration $k(x,y)=\frac{f(y)-f(x)}{y-x}$ is increasing in $y$ on $[x,b]$.

Hence, the right-hand derivative $D_+f(x)$ exists for all $x\in [a,b]$.

In a similar way we conclude that the left-hand derivative $D_-f(x)$ exists for all $x\in[a,b]$, and that $D_-f(x)\leq D_+f(x)$.

Hence the set of points for which is $D_-f(x)< D_+f(x)$ is countable.

If $f$ is convex on $[a,b]$, then then both

$D_+f(x)$ and $D_-f(x)$

are integrable with respect to Lebesgue

measure on

$[a,b]$, and $f(x)=f(a)+ \int_a^x D_+f(t) dt=f(a)+ \int_a^x D_-f(t) dt$.

More generally, suppose D is an increasing, real-valued function defined (at least) on $[a,b)$. Define $g(x) := \int^x_a D(t)dt$, for $a \leq x \leq b$. (Possibly $g(b) =\infty$.) Then $g$ is convex.

Dear @James, as your question is about both application and notion of convex sets, let me bring a new book which was published by the end of 2015.

Geometry of Convex Sets by I. E. Leonard and  J. E. Lewis.

"Geometry of Convex Sets begins with basic definitions of the concepts of vector addition and scalar multiplication and then defines the notion of convexity for subsets of n-dimensional space. Many properties of convex sets can be discovered using just the linear structure. However, for more interesting results, it is necessary to introduce the notion of distance in order to discuss open sets, closed sets, bounded sets, and compact sets. The book illustrates the interplay between these linear and topological concepts, which makes the notion of convexity so interesting.

Thoroughly class-tested, the book discusses topology and convexity in the context of normed linear spaces, specifically with a norm topology on an n-dimensional space..."

http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1119022665.html

Dear @Ljubomir Jacić,

Many thanks for posting the announcement for the book by I.E. Leonard and J.E. Lewis on the geometry of convex sets.   This is a very interesting book and perfectly in keeping with the main thrust of this thread.    And the fact thAT Leonard and Lewis provide a topological space setting for the study of convexity is tremendously important (and useful).

Thanks Ljubomir Jacić   for posting the announcement for the book and  James F Peters. Really notion of convexity  is  fascinating.

I believe that there are people (like , Miroslav Pavlovic,George Stoica Josip Pecaric, Gradimir Milovanovic, Zoran Kadelburg, etc)  interested in inequalities  related to convex functions and therefore  I  give a few inequalities .

In one of my previous answers I mentioned Karamata's inequality (KI)

for convex function. It is interesting that I came to (KI) via

Arslanagic's inequality: If $x + y +z =1, x,y,z \geq 0$, prove that

$\frac{x^3}{x^2 +y +z} + \frac{y^3}{y^2 +x +z} + \frac{z^3}{z^2 +y +x} \geq \frac{1}{7}$.

During Arslanagic communication I get an idea to generalize this result:

Let $X_0=\{ (x,y,z) : x + y +z =1, x,y,z \geq 0 \}$,

$R(x,y,z)=g(x) +g(y)+ g(z)$, and let $g$ be convex function on $[0,1]$. Then

(a) min $R$ on $X_0$ equals $3 g(1/3) $.

Hint: use the supporting line at  the point  $1/3$.

This a special case of Karamata's inequality (what I did not know at that time).

Using supporting lines one can give a simplification of the standard proof. A similar approach has also been given by M. Pavlovic.

We discussed on the seminar  the result obtained in

[MM-MA] Miodrag Mateljevic and Miroljub Albijanic,  Lagrange's Theorem, Convex Functions And Gauss Map, ResearchGate.

Marek Svetlik is taking notes.

In particular,  on Belgrade seminar MM presented the content of paper [MM-MA] and suggested some generalizations of the results   obtained in [MM-MA] to several dimensions. Here are some possibilities:

(S-1) If a surface in R^n  is differentiable at every  point  and the Gauss map is  injective, then   the Gauss map is continuous (see [MM-MA] for the  corresponding result in plane) .

(S-2) Let domain $G$ be homemorphic to 3-dim ball and let $S$ be the boundary of $G$ which is homemorphic to 2-dim sphere $S^2$.

$S$ is a strictly convex surface iff intersection with every plane is empty, a point or a strictly  convex closed curve.

What are appropriate generalization of Theorem 4.1 [MM-MA] in several variables? We give a possibility.

(S-3) Let $f$ be a continuous function defined on a (connected) subdomain $D$ of $R^n$. Then the

function is either strictly convex or strictly concave if and only if every straight line intersects

its graph in at most two points.

For $n=1$, this is Theorem 4.1  [MM-MA].

M. Pavlovic also gave a hint for a proof of (S-3) on RG.

See MM, Note on Convexity and related topics (In preparation), Sec 2, on RG.

@Miodrag Mateljević: Marek Svetlik is taking notes.

Many thanks for your very interesting and helpful postings on convex functions.   You mentioned that Marek Svetlik is taking notes for the seminar on convex functions and related topics.   I hope that it will be possible for you or Marek to post a copy of those notes.

You write: (S-2) Let domain $G$ be homemorphic to 3-dim ball and let $S$ be the boundary of $G$ which is homemorphic to 2-dim sphere $S^2$.

Do you mean "homeomorphic"?