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Shvartsman Pavel - One of the best experts on this subject based on the ideXlab platform.
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On the Core of a Low Dimensional Set-Valued Mapping
2021Co-Authors: Shvartsman PavelAbstract:Let ${\mathfrak M}=({\mathcal M},\rho)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued Mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a ``Finiteness Principle for Lipschitz selections'') provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz Mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this result in two special cases. When $m=2$ we prove it for $X={\bf R}^{2}$, and when $m=1$ we prove it for all choices of $X$. Both of these proofs make use of a simple reiteration formula for the ``core'' of a set-valued Mapping $F$, i.e., for a Mapping $G:{\mathcal M}\to{\mathcal K}_m(X)$ which is Lipschitz with respect to the Hausdorff distance, and such that $G(x)\subset F(x)$ for all $x\in{\mathcal M}$.Comment: 35 pages. arXiv admin note: substantial text overlap with arXiv:2010.0454
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The Core of a 2-Dimensional Set-Valued Mapping. Existence Criteria and Efficient Algorithms for Lipschitz Selections of Low Dimensional Set-Valued Mappings
2021Co-Authors: Shvartsman PavelAbstract:Let ${\mathfrak M}=({\mathcal M},\rho)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued Mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz Mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this result in two special cases. When $m=2$ we prove it for $X={\bf R}^{2}$, and when $m=1$ we prove it for all choices of $X$. Both of these proofs make use of a simple reiteration formula for the "core" of a set-valued Mapping $F$, i.e., for a Mapping $G:{\mathcal M}\to{\mathcal K}_m(X)$ which is Lipschitz with respect to the Hausdorff distance, and such that $G(x)\subset F(x)$ for all $x\in{\mathcal M}$. We also present several constructive criteria for the existence of Lipschitz selections of set-valued Mappings from ${\mathcal M}$ into the family of all closed half-planes in ${\bf R}^{2}$.Comment: 113 pages, 9 figure
Manor Mendel - One of the best experts on this subject based on the ideXlab platform.
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a simple proof of the johnson lindenstrauss extension theorem
American Mathematical Monthly, 2019Co-Authors: Manor MendelAbstract:AbstractJohnson and Lindenstrauss proved that any Lipschitz Mapping from an n-point subset of a metric space into Hilbert space can be extended to the whole space, while increasing the Lipschitz co...
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a simple proof of johnson lindenstrauss extension
arXiv: Metric Geometry, 2018Co-Authors: Manor MendelAbstract:Johnson and Lindenstrauss proved that any Lipschitz Mapping from $n$-point metric space into Hilbert space can be extended while losing at most a factor of $O(\sqrt{\log n})$ in the Lipschitz constant. We present a variation of their argument that avoids dimension reduction and Kirszbraun theorem.
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a simple proof of johnson lindenstrauss extension theorem
arXiv: Metric Geometry, 2018Co-Authors: Manor MendelAbstract:Johnson and Lindenstrauss proved that any Lipschitz Mapping from an $n$-point subset of a metric space into Hilbert space can be extended to the whole space, while increasing the Lipschitz constant by a factor of $O(\sqrt{\log n})$. We present a simplification of their argument that avoids dimension reduction and the Kirszbraun theorem.
Chun Wei - One of the best experts on this subject based on the ideXlab platform.
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quasi lower dimension and quasi Lipschitz Mapping
Fractals, 2017Co-Authors: Haipeng Chen, Chun WeiAbstract:In this paper, we show that the lower dimension is not invariant under quasi-Lipschitz Mapping, and then we find an invariant named the quasi-lower dimension. We also compute the quasi-lower dimension of a class of sets defined by digit restrictions, and then give an example to distinguish the quasi-lower dimension and other dimensions.
Pavel Shvartsman - One of the best experts on this subject based on the ideXlab platform.
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the core of a 2 dimensional set valued Mapping existence criteria and efficient algorithms for Lipschitz selections of low dimensional set valued Mappings
arXiv: Functional Analysis, 2020Co-Authors: Pavel ShvartsmanAbstract:Let ${\mathfrak M}=({\mathcal M},\rho)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued Mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz Mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this result in two special cases. When $m=2$ we prove it for $X={\bf R}^{2}$, and when $m=1$ we prove it for all choices of $X$. Both of these proofs make use of a simple reiteration formula for the "core" of a set-valued Mapping $F$, i.e., for a Mapping $G:{\mathcal M}\to{\mathcal K}_m(X)$ which is Lipschitz with respect to the Hausdorff distance, and such that $G(x)\subset F(x)$ for all $x\in{\mathcal M}$. We also present several constructive criteria for the existence of Lipschitz selections of set-valued Mappings from ${\mathcal M}$ into the family of all closed half-planes in ${\bf R}^{2}$.
Tristan Riviere - One of the best experts on this subject based on the ideXlab platform.
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immersed spheres of finite total curvature into manifolds
Advances in Calculus of Variations, 2014Co-Authors: Andrea Mondino, Tristan RiviereAbstract:We prove that a sequence of possibly branched, weak immersions of the two-sphere S 2 into an arbitrary compact riemannian manifold (M m ,h) with uniformly bounded area and uniformly bounded L 2 −norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz Mapping of S 2 and whose image is made of a connected union of finitely many, possibly branched, weak immersions of S 2 with finite total curvature. We prove moreover that if the sequence belongs to a class γ of π2(M m ) the limiting Lipschitz Mapping of S 2 realizes this class as well. Math. Class. 30C70, 58E15, 58E30, 49Q10, 53A30, 35R01, 35J35, 35J48, 35J50. I Introduction Througout the paper (M m ,h) denotes a connected riemannian manifold and for any x0 ∈ M m we denote respectively by π2(M m ,x0) the homotopy groups of based maps form S 2 into M m sending the south pole to x0 and by π0(C 0 (S 2 ,M m )) the free homotopy classes. It is well known that the group π2(M m ,x0) for different x0's are isomorphic to each other and π2(M) denotes any of the π2(M m ,x0) modulo isomorphisms. Following the classical approach of Douglas and Rado for the Plateau Problem, Sacks and Uhlenbeck proceeded to the minimization of the Dirichlet energy among Mappings ~ of the two sphere S 2 into M m
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immersed spheres of finite total curvature into manifolds
arXiv: Differential Geometry, 2013Co-Authors: Andrea Mondino, Tristan RiviereAbstract:We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz Mapping of $S^2$ and whose image is made of a connected union of finitely many, possibly branched, weak immersions of $S^2$ with finite total curvature. We prove moreover that if the sequence belongs to a class $\gamma$ of $\pi_2(M^m)$ the limiting Lipschitz Mapping of $S^2$ realizes this class as well.