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

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.