Notes about the Caratheodory number

TL;DR

This paper explores conditions under which a compact set in Euclidean space has a Carathéodory number less than n+1, extending classical theorems and introducing new variants of the colorful Carathéodory and Tverberg theorems.

Contribution

It generalizes Fenchel's result on Carathéodory numbers and develops new versions of the colorful Carathéodory theorem and a Tverberg type theorem for convex compact sets.

Findings

Sufficient conditions for reduced Carathéodory number in compacta

Generalized colorful Carathéodory theorem

A new Tverberg type theorem for convex compacta

Abstract

In this paper we give sufficient conditions for a compactum in $\mathbb R^n$ to have Carath\'{e}odory number less than $n+1$, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carath\'{e}odory theorem and give a Tverberg type theorem for families of convex compacta.