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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Equivariant endomorphisms of the space of convex bodies

Author: Rolf Schneider
Journal: Trans. Amer. Math. Soc. 194 (1974), 53-78
MSC: Primary 52A20; Secondary 53C65
MathSciNet review: 0353147
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Abstract: We consider maps of the set of convex bodies in d-dimensional Euclidean space into itself which are linear with respect to Minkowski addition, continuous with respect to Hausdorff metric, and which commute with rigid motions. Examples constructed by means of different methods show that there are various nontrivial maps of this type. The main object of the paper is to find some reasonable additional assumptions which suffice to single out certain special maps, namely suitable combinations of dilatations and reflections, and of rotations if $ d = 2$. For instance, we determine all maps which, besides having the properties mentioned above, commute with affine maps, or are surjective, or preserve the volume. The method of proof consists in an application of spherical harmonics, together with some convexity arguments.

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Keywords: Convex body, Minkowski addition, Hausdorff metric, motion group, equivariant map, support function, mean width, Steiner point, characterization of functionals defined on the set of convex bodies, quermassintegral, convex function, spherical harmonic, multiplier
Article copyright: © Copyright 1974 American Mathematical Society

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