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

Abstract

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.

Complex intersection bodies

“…The situation with complex convex bodies is different, as no systematic studies of these bodies have been carried out, and results appear only occasionally; see for example [31,35,1,42,49,50].…”

A quasi-Funk’s section theorem in n-complex space

“…[8], an extension was achieved as follows: If K is an origin-symmetric star body, L is a star body and for some i such that 0 , i n ∧ ∧ vol ( ) vol ( ) [9] and [10] introduced the complex intersections of star bodies in the complex n -space n C . Koldobsky's solution to complex version of Busemann-Petty problem [10][11][12][13][14][15][16] can be expressed by:…”

How do difference bodies in complex vector spaces look like? A geometrical approach

“…The S 1 -invariant convex bodies are also called R θ -invariant or equilibrated bodies. This class of bodies is well-adapted to the complex structure of the ambient space (see, for instance, [18,19,28]) and shall be used along the paper. Related results concerning convex bodies or valuations in a complex vector space can be found in [4,6,7,18,19,28,33].…”