Intersection Bodies with Certain Symmetries

TL;DR

This paper investigates how specific symmetries influence the geometric properties of convex bodies, focusing on intersection bodies and their sections, with implications for Busemann-Petty problems and hyperplane inequalities.

Contribution

It generalizes the study of convex bodies with complex and quaternionic symmetries by analyzing intersection bodies under block diagonal orthogonal transformations.

Findings

Symmetries significantly affect the properties of intersection bodies.

Results provide new insights into Busemann-Petty type problems.

Hyperplane inequalities are refined for bodies with these symmetries.

Abstract

In this paper we study how certain symmetries of convex bodies affect their geometric properties. In particular, we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations, generalizing complex and quaternionic convex bodies. We conduct a systematic study of sections of bodies with symmetries of this type, with the emphasis on problems of the Busemann-Petty type and hyperplane inequalities. The main role belongs to the class of intersection bodies with symmetries.