Upper bounds for configurations and polytopes in \({\mathbb{R}}^ d\) (Q1086843)

The number of distinct order types of simple numbered configurations of n points in \({\mathbb{R}}^ d\) is shown to be less than \(n^{d(d+1)n}.\) The number of combinatorially distinct labeled simplicial polytopes with n vertices in \({\mathbb{R}}^ d\) is less than \(n^{d(d+1)n}.\) The number of distinct configurations of n points in general position in \({\mathbb{R}}^ d\) is less than \(n^{2d^ 2n}\).