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

Alesker, Semyon

doi:10.1007/s11856-011-0008-6

Cited by 46 publications

(71 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We recall also Alesker's Fourier transform F : Val [8]). In the present paper we will denote the Fourier transform of a valuation φ by (4) φ := Fφ.…”

Section: Valuations and Curvature Measuresmentioning

confidence: 99%

Hermitian integral geometry

2011

View full text Add to dashboard Cite

We give in explicit form the principal kinematic formula for the action of the affine unitary group on C n , together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce also several other canonical bases for the algebra of unitary-invariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone elements.

show abstract

“…We recall also Alesker's Fourier transform F : Val [8]). In the present paper we will denote the Fourier transform of a valuation φ by (4) φ := Fφ.…”

Section: Valuations and Curvature Measuresmentioning

confidence: 99%

Hermitian integral geometry

2011

View full text Add to dashboard Cite

show abstract

“…• For N + 1 ≤ j ≤ N + N p , z j = γ j . In the orthonormal coordinates α j , β j , γ j on E, the boundary of the projection of E is then given by The operations of pull-back and push-forward of translation-invariant valuations under linear maps were defined by Alesker in [6] for continuous valuations, and extended in [14] to the class of Klain-Schneider continuous valuations. Moreover, since pull-back by injection and push-forward by surjection preserve the class of smooth valuations, one obtains by Alesker-Poincaré duality the operations of pushforward by injection and pull-back by surjection between the corresponding spaces of generalized valuations, see e.g.…”

Section: Appendix a Projections Of An O(p) × O(q)-invariant Ellipsoidmentioning

confidence: 99%

Crofton formulas and indefinite signature

Faifman

2017

Geom. Funct. Anal.

View full text Add to dashboard Cite

Abstract. We study the O(p, q)-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an O(p, q)-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the (p − 1) homogeneous case of O(p, p), yielding a Crofton formula for the centro-affine surface area when p ≡ 3 mod 4. We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly for all O(p, 2)-invariant valuations. The proof relies on the exact computation of some integrals of independent interest. Those are related to Selberg's integral and to the Beta function of a matrix argument, except that the positive-definite matrices are replaced with matrices of all signatures. We also analyze the distinguished invariant Crofton distribution supported on the minimal orbit, and show that, somewhat surprisingly, it sometimes defines the trivial valuation, thus producing a distribution in the kernel of the cosine transform of particularly small support. In the heart of the paper lies the description by Muro of the | det X| s family of distributions on the space of symmetric matrices, which we use to construct a family of O(p, q)-invariant Crofton distributions. We conjecture there are no others, which we then prove for O(p, 2) with p even. The functorial properties of Crofton distributions, which serve an important tool in our investigation, are studied by T. Wannerer and the author in the Appendix.

show abstract

“…of even valuations (for the odd case, which is much more involved and will not be needed in this article, see [6]): If φ ∈ Val +,∞ j , then Fφ ∈ Val +,∞ n−j is the valuation with Klain function given by Kl n−j (Fφ) = (Kl j φ) ⊥ .…”

Section: Area Measures and Valuation Theorymentioning

confidence: 99%

“…This not only prompted tremendous progress in integral geometry (see, e.g., [11,13,15,17]) but also aided in the resolution of several old mysteries. For example, the Fourier type transform discovered by Alesker [6],…”

Section: Introductionmentioning

confidence: 99%

Projection functions, area measures and the Alesker–Fourier transform

Dorrek

Schuster

2017

Journal of Functional Analysis

View full text Add to dashboard Cite

Dual to Koldobsky's notion of j-intersection bodies, the class of j-projection bodies is introduced, generalizing Minkowski's classical notion of projection bodies of convex bodies. A Fourier analytic characterization of j-projection bodies in terms of their area measures of order j is obtained. In turn, this yields an equivalent characterization of j-projection bodies involving Alesker's Fourier type transform on translation invariant smooth spherical valuations. As applications of these results, several basic properties of j-projection bodies are established and new non-trivial examples are constructed.

show abstract

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

Cited by 46 publications

References 36 publications

Hermitian integral geometry

Hermitian integral geometry

Crofton formulas and indefinite signature

Projection functions, area measures and the Alesker–Fourier transform

Contact Info

Product

Resources

About