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arXiv · 2202.10116

From harmonic analysis of translation-invariant valuations to geometric inequalities for convex bodies

Jan Kotrbatý·Thomas Wannerer
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Abstract

The Alesker-Bernig-Schuster theorem asserts that each irreducible representation of the special orthogonal group appears with multiplicity at most one as a subrepresentation of the space of continuous translation-invariant valuations with fixed degree of homogeneity. Moreover, the theorem describes in terms of highest weights which irreducible representations appear with multiplicity one. In this paper, we present a refinement of this result, namely the explicit construction of a highest weight vector in each irreducible subrepresentation. We then describe how important natural operations on valuations (pullback, pushforward, Fourier transform, Lefschetz operator, Alesker-Poincaré pairing) act on these highest weight vectors. We use this information to prove the Hodge-Riemann relations for valuations in the case of Euclidean balls as reference bodies. Since special cases of the Hodge-Riemann relations have recently been used to prove new geometric inequalities for convex bodies, our work immediately extends the scope of these inequalities.

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Jan Kotrbatý, Thomas Wannerer. 2022-02-21. From harmonic analysis of translation-invariant valuations to geometric inequalities for convex bodies. https://arxiv.org/abs/2202.10116

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