On empty convex polytopes
For integers \(n\) and \(d\), with \(n > d \geq 2\), the authors examine the smallest integer \(g(n,d)\) such that any set \(S\) of at least \(g(n,d)\) points in general position in \(E^ d\) contains \(n\) points which are the vertices of an empty convex \(d\)-polytope \(P\) (i.e., \(S \cap \text{int} P = \varphi)\). Among other results, they show that if \(g(d + k, d)\) exists, then \(g(d + k,d) \geq d + 2k - 1\) for \(k \geq 1\), i.e. (together with a result of the first named author and V. Soltan) \(g(d + k,d) = d + 2k - 1\) for \(1 \leq k \leq \lfloor {d \over 2} \rfloor + 1\). In this context, the smallest number \(q = q_ t (d)\) of points in \(E^ d\) is determined such that all subsets of size \(q-t\) have a nonempty common intersection.
scientific article; zbMATH DE number 739175
scientific article