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Harmonic Analysis of Translation Invariant Valuations

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Abstract

The decomposition of the space of continuous and translation-invariant valuations into a sum of SO(n) irreducible subspaces is obtained. A reformulation of this result in terms of a Hadwiger-type theorem for continuous translation-invariant and SO(n)-equivariant tensor valuations is also given. As an application, symmetry properties of rigid-motion invariant and homogeneous bivaluations are established and then used to prove new inequalities of Brunn–Minkowski type for convex body valued valuations.

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Authors and Affiliations

  1. School of Mathematical Sciences, Raymond and Beverley Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Semyon Alesker

  2. Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, D-60054, Frankfurt, Germany

    Andreas Bernig

  3. Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstrasse 8–10, 1040, Wien, Austria

    Franz E. Schuster

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  1. Semyon Alesker
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  2. Andreas Bernig
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  3. Franz E. Schuster
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Correspondence to Andreas Bernig.

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Alesker, S., Bernig, A. & Schuster, F.E. Harmonic Analysis of Translation Invariant Valuations. Geom. Funct. Anal. 21, 751–773 (2011). https://doi.org/10.1007/s00039-011-0125-8

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  • DOI: https://doi.org/10.1007/s00039-011-0125-8

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