Fourier Analysis in Convex Geometry

Koldobsky, Alexander

doi:10.1090/surv/116

Cited by 263 publications

(440 citation statements)

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“…In this case, we may consider subspaces E ⊂ R n for which (Proj E ) * µ is ε-radial, and expect that the restriction of · to these subspaces is close, in a certain sense, to the Euclidean norm. See Koldobsky [19,Chapter 6] for a comprehensive discussion of norms admitting representations in the spirit of (46). While this approach may possibly yield some meaningful estimates for some classes of normed spaces, it has limitations.…”

Section: Remarksmentioning

confidence: 99%

“…However, for Banach spaces such as N ∞ , a random subspace is not sufficiently close to a Hilbert space (see Schechtman [28]), and there are better choices than the random one. (Indeed, the N ∞ norm cannot be represented as in (46) or in a similar way, see Theorem 6.13 in Koldobsky [19], due to Misiewicz.) A direct application of Theorem 1.3 is thus quite unlikely to provide new information regarding approximately Hilbertian subspaces for all finite-dimensional normed spaces.…”

Section: Remarksmentioning

confidence: 99%

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On nearly radial marginals of high-dimensional probability measures

Klartag

2010

J. Eur. Math. Soc.

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Section: Remarksmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

On nearly radial marginals of high-dimensional probability measures

Klartag

2010

J. Eur. Math. Soc.

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“…Projection bodies and their polars have received considerable attention over the past few decades (see [3,5,7,8,10,14,15,16,22,25,26,29,32,33,34,35]). Important volume inequalities for the polars of projection bodies are the Petty projection inequality [23] and the Zhang projection inequality [31]: Among bodies of given volume the polar projection bodies have maximal volume precisely for ellipsoids and minimal volume precisely for simplices.…”

Section: Introductionmentioning

confidence: 99%

On Polars of Mixed Complex Projection Bodies

Liu¹,

Wang²,

Huang³

2015

Bulletin of the Korean Mathematical Society

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“…Regarding (1.1), extensive information, including generalizations and applications, can be found in [16,26,36,47,48,49].…”

Section: Introductionmentioning

confidence: 99%

The λ-cosine transforms with odd kernel and the hemispherical transform

Rubin

2014

Fract Calc Appl Anal

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We review some basic facts about the λ-cosine transforms with odd kernel on the unit sphere S n−1 in R n . These transforms are represented by the spherical fractional integrals arising as a result of evaluation of the Fourier transform of homogeneous functions. The related topic is the hemispherical transform which assigns to every finite Borel measure on S n−1 its values for all hemispheres. We revisit the known facts about this transform and obtain new results. In particular, we show that the classical FunkRadon-Helgason inversion method of spherical means is applicable to the hemispherical transform of L p -functions.MSC 2010 : Primary 44A12; Secondary 47G10 Key Words and Phrases: fractional integrals, hemispherical transform, λ-cosine transforms with odd kernel, spherical means IntroductionThe following analytic families of fractional integrals arise as a result of evaluation of the Fourier transform of homogeneous functions on R n \ {0}. Here S n−1 is the unit sphere in R n , λ is a complex parameter, γ n,λ andγ n,λ are normalizing coefficients which will be specified later, d * σ stands for the normalized surface element on S n−1 . We call these operators the λ-cosine transforms with even and odd kernel, respectively. The term cosine transform is borrowed from integral geometry and reflects the fact that θ · σ is nothing but the cosine of the angle between the unit vectors θ and σ. Operators of this kind occur in different branches of mathematics without naming or under different names. In the following, we often skip the prefix λ, for short.The main concern of the present article is the operator family (1.2). Regarding (1.1), extensive information, including generalizations and applications, can be found in [16,26,36,47,48,49].Our next topic is the hemispherical transform which assigns to every finite Borel measure on S n−1 its values for all hemispheres in S n−1 . This transform is intimately connected withC λ . It was introduced by Funk [13] for n = 3 and studied in [4,20,21,43,16,54]. A similar transform for half-spaces in R n was studied in [23,28,41] in connection with Kolmogorov's problem [27, Problem No. 16]. Unlike the half-space transform, the hemispherical transform is non-injective and its kernel is composed by even measures with the mean value zero. The latter means that our consideration can be focused on odd measures or functions.In the present paper we review basic facts about operatorsC λ and the hemispherical transform. We update our previous proofs given in [43] and obtain new results. In particular, in Section 5.4 we show that the Funk-Radon-Helgason inversion method of spherical means, which plays an important role in the theory of Radon transforms [22,50], is applicable to the hemispherical transform.The paper is organized as follows. In Section 2 we recall some facts from Fractional Calculus and analysis on the sphere. In Section 3 we show how (1.1) and (1.2) arise in the framework of the Fourier analysis on R n and discuss properties of these operators, including action in ...

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Fourier Analysis in Convex Geometry

Cited by 263 publications

References 2 publications

On nearly radial marginals of high-dimensional probability measures

On nearly radial marginals of high-dimensional probability measures

On Polars of Mixed Complex Projection Bodies

The λ-cosine transforms with odd kernel and the hemispherical transform

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