Minkowski Valuations

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 Lp-Minkowski valuations. 2000 AMS subject classification: 52A20 (52B11, 52B45) Centroid, difference, and projection bodies are fundamental notions in the affine geometry of convex bodies. The most important affine isoperimetric inequalities (and open problems) are formulated using these bodies. We show that the operators defined by these bodies together with the identity are basically the only examples of homogeneous, SL(n) equivariant or contravariant Minkowski valuations. The centroid body ΓK of a convex body K ⊂ R is a classical notion from geometry (see [5], [16], [36]) that has attracted much attention in recent years (see [4], [6], [8], [20], [21], [25], [27], [31]). If K is o-symmetric, then ΓK is the body whose boundary consists of the locus of the centroids of the halves of K formed when K is cut by hyperplanes through the origin. In general it can be defined in the following way. Let K denote the set of convex bodies (that is, of compact, convex sets) in R, and let K o denote the set of convex bodies in R that contain the origin. A convex body K is uniquely determined by its support function h(K, ·), where h(K, v) = max{v · x : x ∈ K}, v ∈ R, and where v · x denotes the standard inner product of v and x. The moment body MK of K ∈ K o is the convex body whose support function is given by