An invariant property of balls in arrangements of hyperplanes (Q1312192)

The authors give a purely combinatorial and short proof of a result recently obtained by \textit{D. Q. Naiman} ['Hyperplane arrangements and related invariants', unpublished manuscript, January 1992]: Let \(H\) denote a collection of \(n\) hyperplanes in \(d\)-space, where any \(d\) of them meet in exactly one point, any \(d+1\) have empty intersection, and no \(d+2\) are tangent to a common sphere. For each \((d+1)\)-tuple from \(H\), considered as the system of facet hyperplanes of a simplex, take the open ball inscribed in that simplex. If \(B_ k\) denotes the set of such balls intersected by precisely \(k\) hyperplanes \((k=0,1,\dots,n-d-1)\), then \[ | B_ k |={n-k-1 \choose d}. \]