Convolutions and multiplier transformations of convex bodies

Schuster, Franz

doi:10.1090/S0002-9947-07-04270-5

Convolutions and multiplier transformations of convex bodies
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by Franz E. Schuster PDF

Trans. Amer. Math. Soc. 359 (2007), 5567-5591 Request permission

Abstract:

Rotation intertwining maps from the set of convex bodies in $\mathbb {R}^n$ into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator.

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