Valuations and Surface Area Measures

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.