An upper-bound theorem for families of convex sets (Q1073349)

Let \({\mathcal K}\) be a finite family of convex sets in \({\mathbb{R}}^ d\) and let \(f_ k({\mathcal K})\) denote the number of subfamilies of \({\mathcal K}\) of size \(k+1\) with non-empty intersections. The author derives an upper bound for \(f_ k({\mathcal K})\), \(k\leq d+r-1,\) in terms of the size of \({\mathcal K}\) under the assumption that each intersection of \(d+r+1\) members of \({\mathcal K}\) is empty. His geometrical argumentation resembles McMullens proof of the Upper- bound theorem for convex polytopes, as the author remarks.