Sets in [formula omitted]d with no large empty convex subsets

Abstract

Let A be a finite set of points in general position in Rd, d ⩾ 2. Points a1, a2,…, an ϵ A form an n-hole in A (an empty convex subset of A) if they are vertices of a convex polytope containing no other point of A. Let h(d) denote the maximum number h such that any sufficiently large set of points in general position in Rd contains an h-hole. For any d ⩾ 2, we show that 2d + 1 ⩽ h(d) ⩽ 2d−1)(P(d − 1) + 1), where P(d − 1) is the product of the smallest d − 1 prime numbers. For d = 3 we show a better upper bound: h(3) ⩽ 22.