Colorful theorems for strong convexity

TL;DR

This paper extends colorful Carathéodory theorems to strongly convex hulls, providing new theoretical results and examining conditions under which convex bodies can be Minkowski summands.

Contribution

It introduces two new colorful theorems for strongly convex hulls and explores the necessity of the generating convex set assumption.

Findings

Proved two colorful theorems for strongly convex hulls

Analyzed the role of generating convex sets in such theorems

Provided a topological criterion for Minkowski summands

Abstract

We prove two colorful Carath\'eodory theorems for strongly convex hulls, generalizing the colorful Carat\'eodory theorem for ordinary convexity by Imre B\'ar\'any, the non-colorful Carath\'eodory 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.