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

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.