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.

Real-valued valuations on Sobolev spaces

“…Thus, Z : P n 0 → R is a valuation. By Theorem 1, there exist c 0 , c 1 , c 2 ∈ R depending now on s such that (10) z(sℓ P ) = c 0 (s) + c 1 (s) |P | + c 2 (s) |P * | , for all s ∈ R and ℓ P ∈ P 1,p (R n ). We now investigate the behavior of these constants by studying valuations on different sℓ P 's and their translations, for s ∈ R.…”

Moments and valuations

“…Since ν is odd and even, it has to vanish. From (21) we deduce that f I 2 (x, y) = 0. Using (20), (21) and what we have just shown, we see that…”

“…Consequently, ν is odd. From the definition of ν, the SL(2)-covariance of µ for −1 0 0 −1 , the definition of f I , relation (21) and again the definition of ν we infer…”

TheLpMinkowski problem for polytopes for0<p<1

“…Obviously, S 1 (K, ·) is the classical surface area measure of K. In recent years, the L p surface area measure appeared in, e.g., [1,4,8,22,23,25,26,30,[40][41][42][45][46][47]50,51,[53][54][55][56]61]. Today, the L p Minkowski problem is one of the central problems in convex geometric analysis.…”