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half-space cuts

Source: miklos schweitzer 2006 q10

September 3, 2021
convexcompactnessreal analysisEuclidean space

Problem Statement

Let K1,...,KdK_1,...,K_d be convex, compact sets in RdR^d with non-empty interior. Suppose they are strongly separated, which means for any choice of x1∈K1,x2∈K2,...x_1 \in K_1, x_2 \in K_2, ..., their affine hull is a hyperplane in RdR^d. Also let 0<αi<10< \alpha_i <1. A half-space H is called an α\alpha-cut if vol(Ki∩H)=αi⋅vol(Ki)vol(K_i \cap H) = \alpha_i\cdot vol(K_i) for all i. How many α\alpha-cuts are there?
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