I have searched through the internet and found "Convex Polyhedron", "Convex Polytope".
Are they the same thing?
Plus it would be great if there is some elaboration on the properties of such object.
.
strictly speaking, a polyhedron is a bounded 3-dimensional polytope ... whether a polytope is bounded or not depends on the authors but it generalizes the notion of polyhedron to higher dimensions
... however when it comes to naming things, combinatorial geometry sometimes looks like pre-Linne taxonomy ! In the context of convex sets, you'll often encounter a different acception :
- a (convex) polyhedron is the intersection of a finite number of half-spaces (therefore potentially unbounded)
- a (convex) polytope is a bounded (convex) polyhedron (and therefore the convex hull of a finite set of points)
(which is just the reverse of the above regarding the boundedness ...)
http://www.cis.upenn.edu/~jean/combtopol-n.pdf
and a convex polyhedral cone is the convex hull of a finite set of half-lines (therefore obviously unbounded)
http://cowles.econ.yale.edu/P/cm/m13/m13-18.pdf
i am pretty sure that common usage (including mine) has fully mixed all of the above into a hopeless mess, so you'd better define your object of interest by its properties rather than trust its name(s)
.
Thank you so much Fabrice ! But somehow the links no longer work.
I have also just updated my question: Given a convex polyhedral set, now the focus is on the cone spanned by such set. Let's call this cone $K$. Then what does it mean if we say $c_1,...,c_q$ are the vectors generating $K$ ?
Is there any tractable formula to derive them?
.
regarding your last question, i would say that a cone (with vertex at the origin O) spanning a convex polyhedral set S is generated by the set of the vectors OPi (or, in other words, is the convex hull of the half lines starting in O and running through Pi) where Pi are the extreme points of S (and this generating family is not necessarily minimal) BUT this is just from the top of my head (i made no effort to (dis)prove that and i would not be much surprised if it happened to be completely wrong)
.
- Badges
- Science topic
- Similar topics
- Mathematics