Fourier Analysis in Convex Geometry

By Alexander Koldobsky. 2005

387 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/c7ddaf30c5bfbd7b6e403bdfd8604f6d32d0d7f7

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In Applied Mathematics

In Geometry and Topology

In Molecular Biology

Citations in the corpus

OPERATIONS BETWEEN SETS IN GEOMETRY (2012, background): “The analogy between Theorems A and C is yet another instance of a still mysterious, imperfect duality between projections onto subspaces and sections by subspaces, discussed at length in [16] and [26], for example.”
Generalized intersection bodies (2005, methodology): “To this end, we will need the following Spherical Parseval identity, due to Koldobsky ([ Kol05 ]): Spherical Parseval (Koldobsky).”
A characterization of Lp intersection bodies (2006, background): “Valuations allow us to obtain characterizations of many important functionals and operators on convex sets by their invariance or covariance properties with respect to suitable groups of transformations (see [12], [16], [33], [34] for information on the classical theory and [1]–[4], [14], [15],…”
On the stability of Brunn–Minkowski type inequalities (2016, methodology): “For the theory of convex bodies we refer the reader to the books by Ball [1], Bonnesen, Fenchel [5], Koldobsky [20], Milman and Schechtman [30], Schneider [38] and others.”
The dual Minkowski problem for symmetric convex bodies (2017, methodology): “ctions between the dual theory and harmonic analysis. See, for example, Bourgain [5], Gardner [17], Gardner-KoldobskySchlumprecht [19], Lutwak [38], and Zhang [51], and see Gardner [16] and Koldobsky [33] for additional references. Dual curvature measures, parameterized by q ∈ R, are the analogues i”
Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems (2016, positive): “Some of these works bring techniques from harmonic analysis, in particular, Radon transforms and the Fourier transform, into the dual Brunn-Minkowski theory, see [27, 44].”
Optimal Sobolev norms and the Lp Minkowski problem (2006, methodology): “The integral transform used to define Γ−p is the L-cosine transform, which is also the Fourier transform for even homogeneous functions of degree −n − p on R (see Koldobsky [14] for details).”
Adams inequalities on measure spaces (2009, neutral): “the identity Z Rm exp − Xm j=1 x2 j p/2 dx= 2πm/2Γ 1+ m p mΓ m 2 valid for m∈ N and p>0. Note that B−1 n 1 is in fact the volume of the convex body x∈ R4 : P4 1 x 4 j <1 o (see for example [K]). /// 42 6. Sharp Adams inequalities for sums of weighted potentials. As another illustration of how”
Centroid bodies and comparison of volumes (2006, background): “the perturbation argument from [ K5 , p.96]).”