Minkowski Valuations in a 2-Dimensional Complex Vector Space

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 ≥ 3 such a classification was previously established and only valuations of a degree of homogeneity 2m − 1 appear.

Geometric valuation theory

“…Classification results for Minkowski valuations on complex vector spaces were established by Abardia & Bernig [1,2,3]. They introduce and characterize complex projection and difference bodies.…”

Minkowski valuations on lattice polytopes

“…Moreover, (2) and (36) imply that any facet normal of Z T n is a facet normal of Q. Since T n −T n is o-symmetric, (2) implies that it is now sufficient to show that the affine hull of any facet F of [−1, 1] n +[e 1 , .…”

Dual mixed complex brightness integrals

“…with equality if and only if M and N have similar dual complex brightness. Notice that M ⊆ K, N ⊆ L, by (6), (4), (2) and 19, we obtain…”