Convexity spaces. II. Separation

Low complexity functions and convex sets in $\mathbb{Z}^k$

“…In the terminology of [4], the sets S and T in Theorem 2 are a convex pair. The above counterexample shows that in general two disjoint line-convex sets cannot be "separated by a convex pair" (cf.…”

“…The above counterexample shows that in general two disjoint line-convex sets cannot be "separated by a convex pair" (cf. [4,Theorem (3)]). Theorem 2 says, however, that every convex pair can be separated by a hyperplane (cf.…”

“…Theorem 2 says, however, that every convex pair can be separated by a hyperplane (cf. [4,Theorem (6)]).…”

“…It was shown by Bryant and Webster that convexity can be studied from an axiomatic point of view in great generality. In [3], they introduced convexity spaces for this purpose, and in [4] separation in convexity spaces was investigated. A separation theorem corresponding to the one in euclidean spaces or, more generally, convexity spaces does not hold for line-convex sets in lattices.…”

Topological convexity spaces

“…Although the above axioms are algebraic in nature they have a strong geometric motivation based on the vector space example. The properties of joins are discussed in more detail in (2), (5) and some consequences of these axioms are given in (1), (3). We have included here only those properties necessary for our subsequent study of topologies on (X, •).…”