Simultaneous Busemann-Petty and Shephard Volume Comparisons
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≥5, we construct origin-symmetric convex bodies of revolution K,L⊂ ℝn 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≤4, the affirmative solution of the Busemann--Petty problem shows that the section inequalities alone imply |K|≤|L|. Without origin symmetry, we construct such counterexamples in every dimension n≥2, with one body a nontrivial translate of a Euclidean ball and the other a noncentrally symmetric body of constant brightness.
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BibTeX
@misc{zvavitch2026simultaneous,
title = {{Simultaneous Busemann-Petty and Shephard Volume Comparisons}},
author = {Artem Zvavitch},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29505},
eprint = {2608.29505},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29505},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Simultaneous Busemann-Petty and Shephard Volume Comparisons
AU - Artem Zvavitch
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29505
UR - https://arxiv.org/abs/2608.29505
AB - 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≥5, we construct origin-symmetric convex bodies of revolution K,L⊂ 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≤4, the affirmative solution of the Busemann--Petty problem shows that the section inequalities alone imply |K|≤|L|. Without origin symmetry, we construct such counterexamples in every dimension n≥2, with one body a nontrivial translate of a Euclidean ball and the other a noncentrally symmetric body of constant brightness.
ER -