A characterization of the Carath\'eodory number for $H$-convexity

TL;DR

This paper characterizes the Carathéodory number for H-convexity as the maximum of the Helly number and the cone number of H, establishing new bounds and connections with K-strong convexity, and proposing conjectures on its exact value.

Contribution

It introduces a formula for the Carathéodory number for H-convexity, relates it to K-strong convexity, and proposes conjectures on its precise characterization.

Findings

Carathéodory number equals max of Helly number and cone number.

Provides bounds for Carathéodory number in K-strong convexity.

Establishes a lower bound of Carathéodory number by Helly number.

Abstract

We show that the Carath\'eodory number for $H$-convexity is the maximum of two parameters: the Helly number for $H$-convexity and the cone number of $H$. The cone number in this article is defined as the maximal number of points of $H$ in conical position with an empty positive hull relative to the remaining points. Earlier partial results by Boltyanski and Martini can provide an exact value for the Carath\'eodory number only when the Helly number is $1$ or $2$. We further establish connections between the Carath\'eodory numbers for $H$-convexity and that for $K$-strong convexity, where $H$ is the set of normals of $K$. Specifically, the Carath\'eodory number for $H$-convexity provides a lower bound for that of $K$-strong convexity. Moreover, if $K$ is a polytope, which has $|H|$ facets, then the Carath\'eodory number for $K$-strong convexity is at most the maximum of $|H|-1$ and the Carath\'eodory number for $H$-convexity. We conjecture a characterization of when the bound $|H|-1$ is attained. It is a consequence of a broader conjecture stating that the Carath\'eodory number for $K$-strong convexity is at most the maximum of the Carath\'eodory numbers for $H'$-convexity over all subsets $H'\subseteq H$. Finally, we observe that the Carath\'eodory number is at least the Helly number in any convex-structure where all sets are ordinarily convex.