Title: Finite dimensional convex structures I: General results
Abstract: A notion of dimension for topological convex structures has been investigated. It is shown that all inductive dimension functions coincide for convexities with connected convex sets, and that in the latter circumstances dimension can be characterized by means of the dimension of hyperplanes or of convexity preserving maps onto cubes. If the underlying space is separable metric, and if convex sets are even n-connected for all n, then the dimension of the covex structure equals the dimension of the underlying topological space. In contrast. there exits a 0-dimensional separable metric space with a natural ∞-dimensional convexity.