Let K1,...,Kd be convex, compact sets in Rd with non-empty interior. Suppose they are strongly separated, which means for any choice of x1∈K1,x2∈K2,..., their affine hull is a hyperplane in Rd. Also let 0<αi<1. A half-space H is called an α-cut if vol(Ki∩H)=αi⋅vol(Ki) for all i.
How many α-cuts are there? convexcompactnessreal analysisEuclidean space