Projection bodies in complex vector spaces

TL;DR

This paper characterizes a class of Minkowski valuations in complex vector spaces, showing they satisfy key geometric inequalities, thus advancing the understanding of valuation theory in complex geometry.

Contribution

It explicitly describes continuous, translation invariant, contravariant Minkowski valuations in complex vector spaces and proves they satisfy fundamental geometric inequalities.

Findings

Explicit description of Minkowski valuations in complex spaces

Valuations satisfy Brunn-Minkowski, Aleksandrov-Fenchel, Minkowski inequalities

Advances valuation theory in complex geometry

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.