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

The classical Busemann–Petty problem (1956) asks, whether origin-symmetric convex bodies in R n with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n ⩽ 4 and negative if n > 4 . The same question can be asked when volumes of hy...

Full description

Saved in:
Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2010-10, Vol.225 (3), p.1461-1498
Main Author: Rubin, Boris
Format: Article
Language:English
Subjects:
Cosine transforms
Intersection bodies
Quaternions
Spherical Radon transforms
The Busemann–Petty problem
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The classical Busemann–Petty problem (1956) asks, whether origin-symmetric convex bodies in R n with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n ⩽ 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 analysis 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. We discuss possible generalizations.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2010.04.005