SCHWEITZER 2006 Problem 10

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