Sharp uniform convexity and smoothness inequalities for trace norms

Ball, Keith; Cárlen, Eric A.; Lieb, Élliott H.

doi:10.1007/978-3-642-55925-9_18

Cited by 110 publications

(329 citation statements)

References 13 publications

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“…Since ρ X (τ ) ≤ τ , assuming p < q and ρ X (τ ) ≤ Kτ q , where K ≥ 1, we have ρ X (τ ) ≤ Kτ p . By the proof of [3,Proposition 7], it follows that for 1 < p ≤ q, S p (X) ≤ C S q (X) q/p , where C depends only on p, q. The first result now follows from Theorem 2.3.…”

Section: Proofmentioning

confidence: 84%

“…It is straightforward to check that in this case, necessarily, p ≤ 2. By [3,Proposition 7], ρ X (τ ) ≤ Kτ p for every τ > 0 if and only if there exists a constant S > 0 such that for every x, y ∈ X,…”

Section: (Maurey's Extension Theorem)mentioning

confidence: 99%

“…It was shown in [3] (see also [20]) that K 2 (L p ) ≤ 1/ √ p − 1 for 1 < p ≤ 2 and that S 2 (L p ) ≤ √ p − 1 for 2 ≤ p < ∞. (The order of magnitude of these constants was first calculated in [24].…”

Section: (Maurey's Extension Theorem)mentioning

confidence: 99%

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Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces

Peres²,

et al. 2006

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A metric space X has Markov-type 2 if for any reversible finite-state Markov chain {Z t } (with Z 0 chosen according to the stationary distribution) and any map f from the state space to X, the distance, who showed its importance for the Lipschitz extension problem. Until now, however, only Hilbert space (and metric spaces that embed bi-Lipschitzly into it) was known to have Markov-type 2. We show that every Banach space with modulus of smoothness of power-type 2 (in particular, L p for p > 2) has Markov-type 2; this proves a conjecture of Ball (see [2, Section 6]). We also show that trees, hyperbolic groups, and simply connected Riemannian manifolds of pinched negative curvature have Markov-type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in [28, Section 2] by showing that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of L p to L q has a Lipschitz extension defined on all of L p .

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Section: Proofmentioning

confidence: 84%

Section: (Maurey's Extension Theorem)mentioning

confidence: 99%

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Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces

Peres²,

et al. 2006

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“…, where a i,j are the entries of the matrix A. Ky Fan's inequality (see Section 2.11.12 in [40]) demonstrates that Q n |L(S n p ) = 1 but the averaging representation of Q n from [41] implies that Q n is a norm 1 projector from L(l 2 (I n )) endowed with any norm onto its subspace of the diagonal matrixes (with respect to {e k }). In a shortly-formalized form this representation can be written as…”

Section: Non-commutative Spacesmentioning

confidence: 99%

Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces

Ajiev

2013

Axioms

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We introduce and study Clarkson, Dol'nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schäffer constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Hölder-Lipschitz mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative L p -spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the l p -sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration.

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“…Let p be a fixed real number with p ≥ 2. A Banach space E is said to be p-uniformly convex if there exists a constant c > 0 such that δ ( ϵ ) ≥ c ϵ p for all ϵ ∈ [0, 2]; see (Ball et al1994; Takahashi et al 2002) for more details. Observe that every p -uniformly convex is uniformly convex.…”

Section: Preliminariesmentioning

confidence: 99%

The hybrid block iterative algorithm for solving the system of equilibrium problems and variational inequality problems

Saewan

2012

SpringerPlus

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In this paper, we construct the hybrid block iterative algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi - ϕ- asymptotically nonexpansive mappings, the set of the solutions of the variational inequality for an α-inverse-strongly monotone operator, and the set of solutions of a system of equilibrium problems. Moreover, we obtain a strong convergence theorem for the sequence generated by this process in the framework Banach spaces. The results presented in this paper improve and generalize some well-known results in the literature.

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Sharp uniform convexity and smoothness inequalities for trace norms

Cited by 110 publications

References 13 publications

Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces

Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces

Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces

The hybrid block iterative algorithm for solving the system of equilibrium problems and variational inequality problems

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