Projection bodies in complex vector spaces

Abstract

The space of Minkowski valuations on an m-dimensional complex vector space which are continuous, translation invariant and contravariant under the complex special linear group is explicitly described. Each valuation with these properties is shown to satisfy geometric inequalities of Brunn-Minkowski, Aleksandrov-Fenchel and Minkowski type.MSC classification: 52B45, 52A39, 52A40.

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].…”

Difference bodies in complex vector spaces

“…where [e 1 , ie 1 which is smooth only if c = 0. Hence, we get h(ZK, e 1 ) = 0 and from (7) we have h(ZK, αe 1 ) = 0 for all K ∈ K(E) and α ∈ S 1 .…”

Volume inequalities for complex isotropic measures

“…The idea to find analogs of known results from Euclidean geometry in complex vector spaces is not new. In recent years, the study of convex bodies in C n has received considerable attention (see, e.g., [1][2][3][4][5]13,15,19,[22][23][24][25][26]33,34,37,41,42]). …”