A fourier-type transform on translation-invariant valuations on convex sets

Semyon Alesker*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let V be a finite-dimensional real vector space. Let V alsm(V) be the space of translation-invariant smooth valuations on convex compact subsets of V. Let Dens(V) be the space of Lebesgue measures on V. The goal of the article is to construct and study an isomorphism FV: Valsm(V)→̃Valsm(V*) ⊗ Dens(V) such that FV commutes with the natural action of the full linear group on both spaces, sends the product on the source (introduced in [5]) to the convolution on the target (introduced in [16]), and satisfies a Planchereltype formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.

Original languageEnglish
Pages (from-to)189-294
Number of pages106
JournalIsrael Journal of Mathematics
Volume181
Issue number1
DOIs
StatePublished - Mar 2011

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