SL(n) covariant matrix-valued valuations on Lp-spaces

TL;DR

This paper classifies continuous SL(n) covariant matrix-valued valuations on Lp-spaces, revealing their structure and uniqueness, especially highlighting differences in the 2D case involving rotation matrices.

Contribution

It provides a complete classification of such valuations without assuming matrix symmetry, including the unique characterization in higher dimensions and special features in 2D.

Findings

Valuations are characterized by the moment matrix of functions for n>2.

In 2D, rotation matrices play a role in the valuation structure.

The classification is comprehensive and eliminates symmetry assumptions.

Abstract

A complete classification is established for continuous and SL(n) covariant matrix-valued valuations on Lp(Rn,|x|2dx). The assumption of matrix symmetry is eliminated. For n>2, such valuation is uniquely characterized by the moment matrix of measurable function. In the 2-dimensional case, while the rotation matrix shows up.