Centro-Affine Tensor Valuations

TL;DR

This paper classifies all measurable SL(n)-covariant symmetric tensor valuations on convex polytopes containing the origin, showing that they are essentially the moment tensor and a tensor from Lp surface area measures, unifying previous results.

Contribution

It provides a complete classification of such tensor valuations, extending and unifying earlier scalar, vector, and matrix valued results.

Findings

Moment tensor is a fundamental example.

Tensor derived from Lp surface area measures is included.

Classification covers all measurable SL(n)-covariant symmetric tensor valuations.

Abstract

We completely classify all measurable $\operatorname{SL}(n)$-covariant symmetric tensor valuations on convex polytopes containing the origin in their interiors. It is shown that essentially the only examples of such valuations are the moment tensor and a tensor derived from $L_p$ surface area measures. This generalizes and unifies earlier results for the scalar, vector and matrix valued case.