Moments and Valuations

TL;DR

This paper classifies all measurable, SL(n)-covariant vector valuations on convex polytopes containing the origin, showing that the moment vector is essentially the only such valuation.

Contribution

It provides a complete classification of vector valuations with specific covariance properties, identifying the moment vector as the unique valuation of this type.

Findings

Moment vector is the only measurable, SL(n)-covariant vector valuation on convex polytopes.

Complete classification of such valuations under the given conditions.

Establishes the fundamental role of the moment vector in valuation theory.

Abstract

All measurable and $\operatorname{SL}(n)$-covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified. The moment vector is shown to be essentially the only such valuation.