A Fourier type transform on translation invariant valuations on convex sets

Details A Fourier type transform on translation invariant valuations on convex sets A Fourier type transform on translation invariant valuations on convex sets

2007-02-27

arxiv.org/abs/math/0702842v3

Israel J. Math. 181 (2011), 189--294

Mathematics

Metric Geometry

85 pages; revised version; minor corrections

Scientific paper

Let $V$ be a finite dimensional real vector space. In the article we construct an isomorphism between the space of smooth translation invariant valuations on convex subsets of $V$ and the space of such valuations (twisted by densities) on the dual space $V^*$. This isomorphism commutes with the natural action of the full linear group on both spaces of valuations, maps product on the source (introduced in arXiv:math/0301148) to the convolution on the target (introduced in arXiv:math/0607449), and satisfies a Plancherel type inversion formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.

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