A characterization of the Carathéodory number for $H$-convexity

Abstract

We show that the Carathéodory 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 Carathéodory number only when the Helly number is $1$ or $2$. We further establish connections between the Carathéodory numbers for $H$-convexity and that for $K$-strong convexity, where $H$ is the set of normals of $K$. Specifically, the Carathéodory 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 Carathéodory number for $K$-strong convexity is at most the maximum of $|H|-1$ and the Carathéodory 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 Carathéodory number for $K$-strong convexity is at most the maximum of the Carathéodory numbers for $H^{\prime }$-convexity over all subsets $H^{\prime }\subseteq H$. Finally, we observe that the Carathéodory number is at least the Helly number in any convex-structure where all sets are ordinarily convex.

14 pages; comments are welcome