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arXiv · 2302.04347

Inequalities for sections and projections of convex bodies

Apostolos Giannopoulos·Alexander Koldobsky·Artem Zvavitch
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Abstract

This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different volumetric results are considered in a more general setting, with volume replaced by an arbitrary measure. Surprisingly, such a general approach works for a number of volumetric results. In particular, we discuss the Busemann-Petty problem on sections of convex bodies for arbitrary measures and the slicing problem for arbitrary measures. We present generalizations of these questions to the case of functions. A number of generalizations of questions related to projections, such as the problem of Shephard, are also discussed as well as some questions in discrete tomography.

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Apostolos Giannopoulos, Alexander Koldobsky, Artem Zvavitch. 2023-02-08. Inequalities for sections and projections of convex bodies. https://arxiv.org/abs/2302.04347

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