Harmonic analysis of translation invariant valuations

TL;DR

This paper decomposes translation invariant valuations into irreducible components, reformulates a Hadwiger-type theorem for tensor valuations, and applies these results to establish new inequalities for convex body valuations.

Contribution

It provides a new decomposition of valuations and a Hadwiger-type theorem for tensor valuations, advancing the understanding of valuation symmetries and inequalities.

Findings

Decomposition of valuation space into SO(n) irreducible subspaces

Reformulation of Hadwiger theorem for tensor valuations

New Brunn-Minkowski type inequalities for convex body valuations

Abstract

The decomposition of the space of continuous and translation invariant valuations into a sum of SO(n) irreducible subspaces is obtained. A reformulation of this result in terms of a Hadwiger type theorem for continuous translation invariant and SO(n)-equivariant tensor valuations is also given. As an application, symmetry properties of rigid motion invariant and homogeneous bivaluations are established and then used to prove new inequalities of Brunn-Minkowski type for convex body valued valuations.