An invariant property of balls in arrangements of hyperplanes

AbstractLet ℋ be a collection ofn hyperplanes ind-space in general position. For each tuple ofd+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 exactlyk hyperplanes, fork=0, 1,...,n−d−1. We show that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaie% GacaWFcbWaaSbaaSqaaiaa-TgaaeqaaaGccaGLhWUaayjcSdGaeyyp% a0ZaaeWaaeaafaqabeGabaaabaGaamOBaiabgkHiTiaadUgacqGHsi% slcaaIXaaabaGaamizaaaaaiaawIcacaGLPaaaaaa!42DC! $$\left| {B_k } \right| = \left( {\begin{array}{*{20}c} {n - k - 1} \\ d \\ \end{array} } \right)$$ .