Distribution functions of sections and projections of convex bodies

TL;DR

This paper investigates how the distribution of section and projection areas of convex bodies can reveal geometric properties like volume, especially when the directions are unknown, extending classical geometric tomography results.

Contribution

It introduces methods to estimate convex body characteristics from area distribution functions without directional information, advancing geometric tomography techniques.

Findings

Distribution functions can determine volume under certain conditions

New bounds relate area distributions to body size

Results extend classical tomography to unknown directions

Abstract

Typically, when we are given the section (or projection) function of a convex body, it means that in each direction we know the size of the central section (or projection) perpendicular to this direction. Suppose now that we can only get the information about the sizes of sections (or projections), and not about the corresponding directions. In this paper we study to what extent the distribution function of the areas of central sections (or projections) of a convex body can be used to derive some information about the body, its volume, etc.