Minkowski valuations

Abstract

Abstract. Centroid and difference bodies define SL(n) equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of SL(n) equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of SL(n) contravariant Minkowski valuations and of L p -Minkowski valuations.

Minkowski valuations on $L^{p}$-spaces

“…Perhaps the best known result is Hadwiger's classification of continuous rigid motion invariant valuations on K n (see [16,21] and [1,2,3,4,5,6,12,18,19,20,28,37,38] for further results on real-valued valuations). More recently, Minkowski valuations, that is, valuations Z : W → K n , where addition in K n is Minkowski addition, have attracted a lot of interest (see [15,17,23,24,25,26,43,45,46]). Ludwig [25] proved that two important operators on convex bodies can be characterized as Minkowski valuations with certain invariance properties with respect to the special linear group: projection bodies and moment bodies.…”

“…More recently, Minkowski valuations, that is, valuations Z : W → K n , where addition in K n is Minkowski addition, have attracted a lot of interest (see [15,17,23,24,25,26,43,45,46]). Ludwig [25] proved that two important operators on convex bodies can be characterized as Minkowski valuations with certain invariance properties with respect to the special linear group: projection bodies and moment bodies. Whereas projection bodies appear in the classification of valuations on Sobolev spaces [27], this is not the case for moment bodies.…”

“…It turns out that on L p (R n , |x|dx) there are interesting Minkowski valuations, whereas on L p (R n , |x| q dx), where q > 1, it is more natural to consider so-called L q -Minkowski valuations, that is, valuations with respect to L q -Minkowski addition. The study of L q -Minkowski valuations is part of the important L q -Brunn-Minkowski theory (see, for example, [7,8,14,25,31,32,33,34,35,36]), but we will restrict our attention to the classical case q = 1.…”

Asymmetric Blaschke-Santaló functional inequalities

“…The bodies Mε,pK were introduced by Ludwig [12], and afterwards several affine functional inequalities related to the moment and centroid bodies have been extended to the asymmetric case, see for example [27] and [22].…”

“…(see Section 2) and S n,p, . denotes the best Sobolev constant for (12). Following [8], if we choose a specific norm .…”