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Metric cotype
, 2005
"... We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either a ..."
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Cited by 41 (19 self)
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We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either almost universal (i.e., contains any finite metric space with any distortion> 1), or there exists α> 0, and arbitrarily large npoint metrics whose distortion when embedded in any member of F is at least Ω ((log n) α). The same property is also used to prove strong nonembeddability theorems of Lq into Lp, when q> max{2, p}. Finally we use metric cotype to obtain a new type of isoperimetric inequality on the discrete torus. 1
NONLINEAR SPECTRAL CALCULUS AND SUPEREXPANDERS
"... Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesàro averages. Nonlinear spectral gaps of graphs are also shown to behave submultiplicatively under zigzag products. These results yield a ..."
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Cited by 15 (4 self)
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Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesàro averages. Nonlinear spectral gaps of graphs are also shown to behave submultiplicatively under zigzag products. These results yield a combinatorial construction of superexpanders, i.e., a sequence of 3regular graphs that does not admit a coarse embedding into any uniformly convex normed space.
Scaled Enflo Type is Equivalent to Rademacher Type
 Bull. London Math. Soc
"... We introduce the notion of scaled Enflo type of a metric space, and show that for Banach spaces, scaled Enflo type p is equivalent to Rademacher type p. 1 ..."
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Cited by 14 (5 self)
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We introduce the notion of scaled Enflo type of a metric space, and show that for Banach spaces, scaled Enflo type p is equivalent to Rademacher type p. 1
Towards a Calculus for NonLinear Spectral Gaps [Extended Abstract]
"... Given a finite regular graph ..."
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SPECTRAL CALCULUS AND LIPSCHITZ EXTENSION FOR BARYCENTRIC METRIC SPACES
"... The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this biLipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework f ..."
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Cited by 5 (3 self)
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The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this biLipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.
Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
, 2013
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IMPROVED BOUNDS IN THE METRIC COTYPE INEQUALITY FOR BANACH SPACES
"... Abstract. It is shown that if (X, ‖·‖X) is a Banach space with Rademacher cotype q then 1 1+ for every integer n there exists an even integer m � n q such that for every f: Zn m → X we have n∑ ∥∥∥f ( Ex x + m ..."
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Cited by 4 (4 self)
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Abstract. It is shown that if (X, ‖·‖X) is a Banach space with Rademacher cotype q then 1 1+ for every integer n there exists an even integer m � n q such that for every f: Zn m → X we have n∑ ∥∥∥f ( Ex x + m
RADEMACHER AVERAGES ON NONCOMMUTATIVE SYMMETRIC SPACES
, 803
"... Abstract. Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let (εk)k≥1 be a Rademacher sequence, on some probability space Ω. For finite sequences (xk)k≥1 of E(M), we co ..."
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Cited by 2 (0 self)
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Abstract. Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let (εk)k≥1 be a Rademacher sequence, on some probability space Ω. For finite sequences (xk)k≥1 of E(M), we consider the Rademacher averages ∑ k εk ⊗ xk as elements of the noncommutative function space E(L ∞ (Ω)⊗M) and study estimates for their norms ‖ ∑ k εk ⊗ xk‖E calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2concave, ‖ ∑ k εk⊗xk‖E is equivalent to the infimum of ‖ ( ∑ y ∗ 1 kyk) 2 ‖+‖ ( ∑ zkz ∗ k)1 2 ‖ over all yk, zk in E(M) such that xk = yk+zk for any k ≥ 1. Dual estimates are given when E is 2convex and has a non trivial upper Boyd index. In this case, ‖ ∑ k εk ⊗ xk‖E is equivalent to ‖ ( ∑ x ∗ 1 kxk) 2 ‖ + ‖ ( ∑ xkx ∗ k)12 ‖. We also study Rademacher averages ∑ i,j εi ⊗ εj ⊗ xij for doubly indexed families (xij)i,j of