Sets in \(\mathbb{R}^ d\) with no large empty convex subsets
Let \(A\) be a finite set of points in general position in \(\mathbb{R}^ d\), \(d\geq 2\). A subset \(B\subseteq A\) of \(n\) points is called an \(n\)-hole in \(A\) if the points in \(B\) are the vertices of a convex polytope which contains no other point of \(A\). Let \(h(d)\) be the maximum number \(n\) such that any sufficiently large set of points in general position in \(\mathbb{R}^ d\) contains an \(n\)-hole. The author shows that \[ 2d+1\leq h(d)\leq 2^{d-1}(P(d-1)+1), \] where \(P(d-1)\) denotes the product of the smallest \(d-1\) prime numbers. He also gives an improved upper bound in \(\mathbb{R}^ 3\), namely \(h(3)\leq 22\).
scientific article; zbMATH DE number 97492