Methods of Geometric Analysis in Extension and Trace Problems

Brudnyi, Alexander; Brudnyi, Yuri

doi:10.1007/978-3-0348-0209-3

Cited by 56 publications

(67 citation statements)

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“…inequalities are among the main tools of this approach. The results presented in [4] deal with the technically simplest case of Morrey-Campanato spaces (more general results will appear in the forthcoming book [5]). …”

Section: 1mentioning

confidence: 97%

On covering numbers of sublevel sets of analytic functions

Brudnyi

2010

Journal of Approximation Theory

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Section: 1mentioning

confidence: 97%

On covering numbers of sublevel sets of analytic functions

Brudnyi

2010

Journal of Approximation Theory

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“…which is a continuity property similar to (8) with an improved bound. To see that this inequality is sharp, take k = 2, a 1 = a 2 = 1/2 and denote m = bar…”

Section: Continuity Properties Of Convex Combinationsmentioning

confidence: 96%

Borsuk–Dugundji type extension theorems with Busemann convex target spaces

Espínola¹,

Castro²,

Nicolae³

2018

Ann. Acad. Sci. Fenn. Math.

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“…Such questions on "smooth extrapolation from partial data" have been intensively studied over the past century, and a variety of deep results and applications have been obtained, in addition to the development of powerful techniques for the purpose of Lipschitz extension that subsequently became useful in other areas. See the monograph [BB12] for a (very small) sample of such results.…”

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confidence: 99%

Probabilistic clustering of high dimensional norms

Naor

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Separating decompositions of metric spaces are an important randomized clustering paradigm that was formulated by Bartal in [Bar96] and is defined as follows. Given a metric space (X, dX ), its modulus of separated decomposability, denoted SEP(X, dX ), is the infimum over those σ ∈ (0, ∞] such that for every finite subset S ⊆ X and every ∆ > 0 there exists a distribution over random partitions P of S into sets of diameter at most ∆ such that for every x, y ∈ S the probability that both x and y do not fall into the same cluster of the random partition P is at most σdX (x, y)/∆.Here we obtain new bounds on SEP(X, · X ) when (X, · X ) is a finite dimensional normed space, yielding, as a special Our new bounds on the modulus of separated decomposability rely on extremal results for orthogonal hyperplane projections of convex bodies, specifically using the work [BN02] of Barthe and the author. This yields additional refined estimates, an example of which is that for every n ∈ N and k ∈ {1, . . . , n} we have SEP ( n 2 ) k k log(en/k), where ( n 2 ) k denotes the subset of R n consisting of all those vectors that have at most k nonzero entries, equipped with the Euclidean metric.The above statements have implications to the Lipschitz extension problem through its connection to random partitions that was developed by Lee and the author in [LN04,LN05]. Given a metric space (X, dX ), let e(X) denote the infimum over those K ∈ (0, ∞] such that for every * Supported by BSF grant 2010021, the Packard Foundation and the Simons Foundation. The research that is presented here was conducted under the auspices of the Simons Algorithms and Geometry (A&G) Think Tank.† A full version of this extended abstract, titled "Separating decompositions in high dimensions, extremal hyperplane projections, and Lipschitz extension" is forthcoming. It contains several additional results and extensions of the work that is presented here, but also this extended abstract covers some material that does not appear in the full version.

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Methods of Geometric Analysis in Extension and Trace Problems

Cited by 56 publications

References 0 publications

On covering numbers of sublevel sets of analytic functions

On covering numbers of sublevel sets of analytic functions

Borsuk–Dugundji type extension theorems with Busemann convex target spaces

Probabilistic clustering of high dimensional norms

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