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arXiv · 2608.29505

Simultaneous Busemann-Petty and Shephard Volume Comparisons

Artem Zvavitch
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Abstract

We study the simultaneous Busemann-Petty and Shephard volume comparison problem: whether comparison of the volumes of all central hyperplane sections and all orthogonal hyperplane projections determines the ordering of the volumes of two convex bodies. For every $n\geq5$, we construct origin-symmetric convex bodies of revolution $K,L\subset {\mathbb R}^n$ such that every central hyperplane section and every orthogonal hyperplane projection of $K$ has strictly smaller volume than the corresponding section or projection of $L$, while $|K|>|L|$. For $n\leq4$, the affirmative solution of the Busemann--Petty problem shows that the section inequalities alone imply $|K|\leq|L|$. Without origin symmetry, we construct such counterexamples in every dimension $n\geq2$, with one body a nontrivial translate of a Euclidean ball and the other a noncentrally symmetric body of constant brightness.

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Artem Zvavitch. 2026-08-30. Simultaneous Busemann-Petty and Shephard Volume Comparisons. https://arxiv.org/abs/2608.29505

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