  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.

Metric Geometry