Inequalities for sections and projections of convex bodies

TL;DR

This paper explores generalized volumetric inequalities in geometric tomography, extending classical problems like Busemann-Petty and slicing to arbitrary measures and functions, revealing broader applicability of these geometric concepts.

Contribution

It introduces a new approach to geometric tomography by replacing volume with arbitrary measures, generalizing key problems like Busemann-Petty and slicing to this broader setting.

Findings

Generalized Busemann-Petty problem for arbitrary measures

Extended slicing problem to functions and measures

Discussed generalizations of projection-related questions

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.