Document Type
Article
Publication Date
10-1-2010
Abstract
The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in Rn 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. © 2010 Elsevier Inc.
Publication Source (Journal or Book title)
Advances in Mathematics
First Page
1461
Last Page
1498
Recommended Citation
Rubin, B. (2010). Comparison of volumes of convex bodies in real, complex, and quaternionic spaces. Advances in Mathematics, 225 (3), 1461-1498. https://doi.org/10.1016/j.aim.2010.04.005