Abstract
For (Formula presented.), let S be a set of points in (Formula presented.) in general position. A set I of k points from S is a k-island in S if the convex hull (Formula presented.) of I satisfies (Formula presented.).
A k-island in S in convex position is a k-hole in S. For (Formula presented.) and a convex body (Formula presented.) of volume 1, let S be a set of n points chosen uniformly and independently at random from K.
We show that the expected number of k-holes in S is in (Formula presented.). Our estimate improves and generalizes all previous bounds.
In particular, we estimate the expected number of empty simplices in S by (Formula presented.). This is tight in the plane up to a lower-order term.
Our method gives an asymptotically tight upper bound (Formula presented.) even in the much more general setting, where we estimate the expected number of k-islands in S.