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Variants of the Busemann-Petty problem and of the Shephard problem. (English) Zbl 1405.52008

Summary: We provide an affirmative answer to a variant of the Busemann-Petty problem, proposed by V. Milman: Let \(K\) be a convex body in \(\mathbb R^n\) and let \(D\) be a compact subset of \(\mathbb R^n\) such that, for some \(1\leqslant k\leqslant n-1\), \[ | P_F (K)|\leqslant| D\cap F| \] for all \(F\in G_{n,k}\), where \(P_F(K)\) is the orthogonal projection of \(K\) on to \(F\) and \(D\cap F\) is the intersection of \(D\) with \(F\). Then, \[ | K|\leqslant| D|. \] We also provide estimates for the lower dimensional Busemann-Petty and Shephard problems, and we prove separation in the original Busemann-Petty problem.

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
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