Tight Bounds on the Expected Number of Holes in Random Point Sets

For integers \(d\ge 2\) and \(k\ge d+1\), a k-hole in a set S of points in general position in \(\mathbb {R}^d\) is a k-tuple of points from S in convex position such that the interior of their convex hull does not contain any point from S. For a convex body \(K\subseteq \mathbb {R}^d\) of unit volume, we study the expected number \(EH^K_{d,k}(n)\) of k-holes in a set of n points drawn uniformly and independently at random from K.