This is an announcement for the paper "Comparison of volumes of convex bodies in real, complex, and quaternionic spaces" by Boris Rubin.

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 more general comparison functions. 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 for spaces over Clifford algebras are discussed.

Archive classification: math.FA

Mathematics Subject Classification: 44A12; 52A38

Remarks: 38 pages

The source file(s), quaternion3.tex: 107627 bytes, is(are) stored in gzipped form as 0812.1300.gz with size 35kb. The corresponding postcript file has gzipped size 182kb.

Submitted from: borisr@math.lsu.edu

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