On the Carathéodory number of interval and graph convexities

Inspired by a result of Carathéodory [Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911) 193–217], the Carathéodory number of a convexity space is defined as the smallest integer k such that for every subset U of the ground set V and every element u in the convex hull of U, there is a subset F of U with at most k elements such that u in the convex hull of F. We study the Carathéodory number for generalized interval convexities and for convexity spaces derived from finite graphs. We establish structural properties, bounds, and hardness results.