A cogenerator for preseparated superconvex spaces (Q2563762)

A convex space is a set \(X\) for which (finite) convex combinations of points are defined. If convex combinations with countably many terms are also defined, the space is called superconvex. A superconvex space is preseparated if for all \(x,y,z\in X\), \(\alpha\in] 0,1[\), \(\alpha x+(1-\alpha) z=\alpha y+ (1-\alpha)z\) implies \(x=y\). The authors prove that the unit interval \([0,1]\), with its canonical superconvex structure, is a cogenerator for the category of preseparated superconvex spaces. A convex space \(X\) is said to be separated if the family of all affine mappings \(X\to[0,1]\) is point-separating. Separated convex spaces are characterized and it is proved that each of them can be embedded into the unit ball of a real normed vector space.