Tight bounds on the expected number of holes in random point sets

Abstract

AbstractFor integers and , a ‐hole in a set of points in general position in is a ‐tuple of points from in convex position such that the interior of their convex hull does not contain any point from . For a convex body of unit ‐dimensional volume, we study the expected number of ‐holes in a set of points drawn uniformly and independently at random from . We prove an asymptotically tight lower bound on by showing that, for all fixed integers and , the number is at least . For some small holes, we even determine the leading constant exactly. We improve the currently best‐known lower bound on by Reitzner and Temesvari (2019). In the plane, we show that the constant is independent of for every fixed and we compute it exactly for , improving earlier estimates by Fabila‐Monroy, Huemer, and Mitsche and by the authors.