Minkowski valuations in a 2-dimensional complex vector space

TL;DR

This paper classifies continuous, translation invariant Minkowski valuations in a 2D complex vector space that are covariant or contravariant under the complex special linear group, revealing a unique sum of valuations of degrees 1 and 3.

Contribution

It provides the first complete classification of such valuations in 2D complex spaces, extending previous results from higher dimensions.

Findings

Valuations are sums of degrees 1 and 3 in 2D complex space

Classification differs from higher dimensions where only degree 2m-1 valuations appear

Establishes a foundational understanding of Minkowski valuations in complex vector spaces

Abstract

The classification of continuous, translation invariant Minkowski valuations which are contravariant (or covariant) with respect to the complex special linear group is established in a 2-dimensional complex vector space. Every such valuation is given by the sum of a valuation of degree of homogeneity 1 and 3. In dimensions $m\geq 3$ such a classification was previously established and only valuations of a degree of homogeneity 2m-1 appear.