Valuations and Surface Area Measures

TL;DR

This paper characterizes surface area measures and their $L_p$ variants as unique valuations with specific covariance properties, linking classical and $L_p$ Brunn-Minkowski theory.

Contribution

It establishes the uniqueness of the classical surface area measure as an SL(n) contravariant valuation and characterizes $L_p$ surface area measures for all real p.

Findings

Classical surface area measure is essentially unique under SL(n) contravariance.

$L_p$ surface area measures are characterized for all real p.

Provides a unifying framework connecting classical and $L_p$ valuation theories.

Abstract

We consider valuations defined on polytopes containing the origin which have measures on the sphere as values. We show that the classical surface area measure is essentially the only such valuation which is SL(n) contravariant of degree one. Moreover, for all real $p$, an $L_p$ version of the above result is established for GL(n) contravariant valuations of degree $p$. This provides a characterization of the $L_p$ surface area measures from the $L_p$ Brunn-Minkowski theory.