An invariant property of balls in arrangements of hyperplanes

Abstract

Let ℋ be a collection of n hyperplanes in d-space in general position. For each tuple of d+1 hyperplanes of ℋ consider the open ball inscribed in the simplex that they form. Let ℬk denote the number of such balls intersected by exactly k hyperplanes, for k=0, 1,..., n-d-1. We show that {Mathematical expression}. © 1993 Springer-Verlag New York Inc.