On the Carath\'eodory number for strong convexity

TL;DR

This paper improves the Carathéodory theorem for strong convexity in real n-dimensional space, establishing that the Carathéodory number can be reduced to n in certain cases and proving it cannot be smaller than n generally.

Contribution

It provides an improved bound for the Carathéodory number in strong convexity and introduces a new topological criterion for Minkowski summands.

Findings

Carathéodory number reduced to n in several cases

Carathéodory number cannot be smaller than n for any gauge body

New topological criterion for Minkowski summands

Abstract

We give an improvement of the Carath\'eodory theorem for strong convexity (ball convexity) in $\mathbb R^n$, reducing the Carath\'eodory number to $n$ in several cases; and show that the Carath\'eodory number cannot be smaller than $n$ for an arbitrary gauge body $K$. We also give an improved topological criterion for one convex body to be a Minkowski summand of another.