Variants of the Busemann-Petty problem and of the Shephard problem

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We provide an affirmative answer to a variant of the Busemann-Petty problem, proposed by V. Milman: Let K be a convex body in Rⁿ and let D be a compact subset of Rⁿ such that, for some 1 k n-1, |PF(K)| |D∩ F| for all F∈ Gₙ,ₖ, where PF(K) is the orthogonal projection of K onto F and D∩ F is the intersection of D with F. Then, |K| |D|. We also provide estimates for the lower dimensional Busemann-Petty and Shephard problems, and we prove separation in the original Busemann-Petty problem.