Blaschke- and Minkowski-endomorphisms of convex bodies

Abstract

Abstract. We consider maps of the family of convex bodies in Euclidean ddimensional space into itself that are compatible with certain structures on this family: A Minkowski-endomorphism is a continuous, Minkowski-additive map that commutes with rotations. For d ≥ 3, a representation theorem for such maps is given, showing that they are mixtures of certain prototypes. These prototypes are obtained by applying the generalized spherical Radon transform to support functions. We give a complete characterization of weakly monotonic Minkowski-endomorphisms. A corresponding theory is developed for Blaschke-endomorphisms, where additivity is now understood with respect to Blaschke-addition. Using a special mixed volume, an adjoining operator can be introduced. This operator allows one to identify the class of Blaschke-endomorphisms with the class of weakly monotonic, non-degenerate and translation-covariant Minkowski-endomorphisms.The following application is also shown: If a (weakly monotonic and) nontrivial endomorphism maps a convex body to a homothet of itself, then this body must be a ball.

Minkowski valuations

“…He obtained a complete classification of endomorphisms in K 2 and characterizations of special endomorphisms in K n . These results were further extended by Kiderlen [10]. We show that the moment and difference operators are basically the only examples of homogeneous, SL(n) equivariant Minkowski valuations.…”

VOLUME INEQUALITIES FOR THE Lp-SINE TRANSFORM OF ISOTROPIC MEASURES

“…However, in contrast to L p projection bodies, the operators we consider will not be compatible with general linear transformations but merely with rotations (see [10,12,13,21,33,37,40,42] for recent results concerning such operators). Therefore we will put the convex bodies in L p surface isotropic position, a notion that we will recall in the following.…”

On polars of Blaschke-Minkowski homomorphisms

“…Blaschke-Minkowski homomorphism is an important notion in the theory of convex body valued valuations (see, e.g., [1], [5], [8], [10], [12]- [14], [17], [21], [23]- [25], [30]). Their natural dual, radial Blaschke-Minkowski homomorphism, was introduced by Schuster [20] and further investigated to be meaningful (see [22]).…”