Blaschke- and Minkowski-endomorphisms of convex bodies

Kiderlen, Markus

doi:10.1090/S0002-9947-06-03914-6

Blaschke- and Minkowski-endomorphisms of convex bodies
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Trans. Amer. Math. Soc. 358 (2006), 5539-5564 Request permission

Abstract:

We consider maps of the family of convex bodies in Euclidean $d$-dimensional 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 $d\ge 3$, 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) non-trivial endomorphism maps a convex body to a homothet of itself, then this body must be a ball.

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