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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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