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arXiv · 1207.7286

Rotation invariant Minkowski classes of convex bodies

Rolf Schneider·Franz E. Schuster
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Abstract

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T1; T2 such that M + T1 = T2, and T1; T2 belong to the rotation invariant Minkowski class generated by K. It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K, which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof.

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Rolf Schneider, Franz E. Schuster. 2012-07-31. Rotation invariant Minkowski classes of convex bodies. https://arxiv.org/abs/1207.7286

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