Comparison of volumes of convex bodies in real, complex, and quaternionic spaces

TL;DR

This paper unifies the study of volume comparison problems for convex bodies across real, complex, and quaternionic spaces, revealing dimension-dependent affirmative answers using cosine transforms.

Contribution

It provides a unified treatment of the Busemann-Petty problem in different spaces and characterizes when the problem has an affirmative answer in quaternionic spaces.

Findings

Busemann-Petty problem is affirmative for n ≤ 4 in real spaces.

In quaternionic spaces, the problem is affirmative only when n=2.

The approach uses properties of cosine transforms on the sphere.

Abstract

The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in $\mathbb {R}^n$ with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if $n\le 4$ and negative if $n>4$. The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified exposition of this circle of problems in real, complex, and quaternionic $n$-dimensional spaces. All cases are treated simultaneously. In particular, we show that the Busemann-Petty problem in the quaternionic $n$-dimensional space has an affirmative answer if and only if $n =2$. The method relies on the properties of cosine transforms on the unit sphere. Possible generalizations are discussed.