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

By Semyon Alesker. 2007

75 citations recorded by the source corpus.

AbstractLet 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 $$ \mathbb{F}_V :Val^{sm} (V)\tilde \to Val^{sm} (V^* ) \otimes Dens(V) $$ such that $$ \mathbb{F}_V $$ 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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d02ec337d102d1aacd4a9f93378e8c56f3b639de

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Kinematic formulas for tensor valuations (2014, methodology): “The picture is completed by a Fourier-type transform, due to Alesker [7, 11], which is a linear map F : Val(V ) → Val(V ∗)⊗Dens(V ) satisfying F(φ1 · φ2) = F(φ1) ∗ F(φ2) (2) and with F(2) := F ◦ F also the Plancherel-type formula (F(2)(φ))(K) = φ(−K), K ∈ K(V ).”
Difference bodies in complex vector spaces (2012, background): “Some of the most important and recent results on the theory of continuous translation invariant valuations can be found in [3, 7, 23, 34, 36].”
A Hadwiger-Type Theorem for the Special Unitary Group (2008, background): “The theory of translation invariant continuous valuations is a very rich one, see for instance [22], [1], [2], [20], [13], [8].”
Projection bodies in complex vector spaces (2011, background): “e-dimensional real vector space of dimension n. The Banach space of real-valued, translation invariant, continuous valuations on V is denoted by Val. It has been studied intensively, see for instance [1, 2, 20, 38]. A basic structure result for Val is the following direct sum decomposition due to Mc”
Minkowski valuations and generalized valuations (2015, background): “Alesker [8] also proved that the Poincaré duality map admits a unique continuous extension to the space Vali of continuous translation invariant valuations of degree i.”
Hermitian integral geometry (2008, methodology): “In this form, the Alesker-Fourier transform was denoted by D in [1], [13] and in several other papers.”
Harmonic Analysis of Translation Invariant Valuations (2010, methodology): “The description of smooth valuations provided by Lemma 2.3 was the main tool used in [BB] to establish a hard Lefschetz theorem for translation-invariant valuations (see also [Al4, Al7 ]).”
Algebraic Integral Geometry (2010, background): “Equation (20) in the odd case was established in [15].”