COLORFUL THEOREMS FOR STRONG CONVEXITY

Abstract

We prove two colorful Caratheodory theorems for strongly convex hulls, generalizing the colorful Caratheodory theorem for ordinary convexity by Imre Barany, the non-colorful Caratheodory theorem for strongly convex hulls by the second author, and the "very colorful theorems" by the first author and others. We also investigate if the assumption of a "generating convex set" is really needed in such results and try to give a topological criterion for one convex body to be a Minkowski summand of another.