Geodesic
Metrizability of Finite Dimensional Spaces with a Binary Convexity
Canadian journal of mathematics, Tome 38 (1986) no. 1, pp. 1-18

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A convex structure consists of a set X, together with a collection of subsets of X, which is closed under intersection and under updirected union. The members of are called convex sets, and is a convexity on X. Fox A a subset of X, h (A) denotes the (convex) hull of A. If A is finite, then h(A) is called a polytope, is called a binary convexity if each finite collection of pairwise intersecting convex sets has a nonempty intersection. See [8], [21] for general references.If X is also equipped with a topology, then the corresponding weak topology is the one generated by the convex closed sets. It is usually assumed that at least all polytopes are closed. is called normal provided that for each two disjoint convex closed sets C, D there exist convex closed sets C′, D′, with 
Vel, M. Van de. Metrizability of Finite Dimensional Spaces with a Binary Convexity. Canadian journal of mathematics, Tome 38 (1986) no. 1, pp. 1-18. doi: 10.4153/CJM-1986-001-3
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