I'd like to have a definition of convex set not seen as a subset of an affine or vectorial space but as an abstract set.
The fact that an abstract convex space is isomorphic to a convex subspace of a vectorial space will be, under certain hypothesis, a theorem for the theory of abstract convex spaces.
Hi,
I here attach you some recent paper about abstract convex sets. I hope they will be useful to you.
http://content.schweitzer-online.de/static/catalog_manager/live/media_files/representation/zd_std_orig__zd_schw_orig/004/942/724/9781107601017_foreword_pdf_1.pdf
https://books.google.es/books?hl=es&lr=&id=87XbBwAAQBAJ&oi=fnd&pg=PA1&dq=Abstract+convex+sets&ots=L5xmkOpZFh&sig=UDCc7ImriolN8BUTHGfwXJGa5dM#v=onepage&q=Abstract%20convex%20sets&f=false
You may see:
[1] Bryant, V. W. Independent axioms for convexity. J. of Geometry, 5, 1974, 95-99
[2]n Bryant, V.W. and Webster R.J. Convexity spaces I: J. Math.Anal.Appl. 37, 1972, 206-215, II (the same journal) 1973, 321-327; III 57 1977, 382-392
I think you need to be more specific.
In the theory of generalized ordered sets, convexity is well known (at least in papers).
In a set S, if we have a ternary betweenness relation R defined then a subset K of S is convex iff K is closed under betweenness.
i.e if Rxyz means y is between x and z, then Rxyz for x, z in K implies y in K is the convex closure property..
In the ordered set approach, there are results relating these concepts to vector spaces
and matroids but then few strong axioms are involved :-)
There is an abstract/axiomatic notion of convexity, called "order structure" that essentially goes back to Pasch 1882. If one carefully pushes the theory one can prove a "fundamental theorem" describing when an abstract convex structure can be (uniquely) realized as convex region in Rn.
This is developed in the book : W. A. Coppel, Foundations of convex geometry. Austral. Math. Soc. Lect. Ser. 12, Cambridge University Press, 1998
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