Metric Inequalities

“…The above discretization problem was introduced in [13] as an alternative (quantitative) approach to an important rigidity theorem of Ribe [82]. Additional applications to embedding theory appear in [68,39,69]. To date, the bound (2) remains the best known, even under the additional restriction that Y is uniformly convex.…”

Section: Amentioning

confidence: 99%

Heat flow and quantitative differentiation

Hytönen

2019

J. Eur. Math. Soc.

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For every Banach space (Y, · Y ) that admits an equivalent uniformly convex norm we prove that there exists c = c(Y ) ∈ (0, ∞) with the following property. Suppose that n ∈ N and that X is an n-dimensional normed space with unit ball BX . Then for every 1-Lipschitz function f : BX → Y and for every ε ∈ (0, 1/2] there exists a radius r exp(−1/ε cn ), a point x ∈ BX with x + rBX ⊆ BX , and an affine mapping Λ :εr for every y ∈ x+rBX. This is an improved bound for a fundamental quantitative differentiation problem that was formulated by Bates, Johnson, Lindenstrauss, Preiss and Schechtman (1999), and consequently it yields a new proof of Bourgain's discretization theorem (1987) for uniformly convex targets. The strategy of our proof is inspired by Bourgain's original approach to the discretization problem, which takes the affine mapping Λ to be the first order Taylor polynomial of a time-t Poisson evolute of an extension of f to all of X and argues that, under appropriate assumptions on f , there must exist a time t ∈ (0, ∞) at which Λ is (quantitatively) invertible. However, in the present context we desire a more stringent conclusion, namely that Λ well-approximates f on a macroscopically large ball, in which case we show that for our argument to work one cannot use the Poisson semigroup. Nevertheless, our strategy does succeed with the Poisson semigroup replaced by the heat semigroup. As a crucial step of our proof, we establish a new uniformly convex-valued Littlewood-Paley-Stein G-function inequality for the heat semigroup; influential work of Martínez, Torrea and Xu (2006) obtained such an inequality for subordinated Poisson semigroups but left the important case of the heat semigroup open. As a byproduct, our proof also yields a new and simple approach to the classical Dorronsoro theorem (1985) even for real-valued functions.

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“…Beyond metric cotype, dimension-dependent scaling parameters occur (for conceptually distinct reasons) in other metric inequalities that arise in the Ribe program. Determining their asymptotically sharp values is a major difficulty that pertains to important open problems; see [120,61,176,60,135,130,137,48]. The currently best-known general bound [60] for the metric cotype q scaling parameter of Banach spaces of Rademacher cotype q is m n 1+1/q .…”

Section: Theorem 2 There Exists a Metric Space Z That Does Not Embedmentioning

confidence: 99%

Nonpositive curvature is not coarsely universal

Eskenazis

Mendel

2019

Invent. math.

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We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni, Naor and Neiman (2015). We establish this statement by proving that a metric space which is q-barycentric for some q ∈ [1,∞) has metric cotype q with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the ℓ ∞ grid [m] n ∞ = ({1,... ,m} n , · ∞ ) into ℓ q for all q ∈ (2,∞), from which we deduce the following discrete converse to the fact that ℓ n ∞ embeds with distortion O(1) into ℓ q for q = O(logn). A rigidity theorem of Ribe (1976) implies that for every n ∈ N there exists m ∈ N such that if [m] n ∞ embeds into ℓ q with distortion O(1), then q is necessarily at least a universal constant multiple of log n. Ribe's theorem does not give an explicit upper bound on this m, but by the work of Bourgain (1987) it suffices to take m = n, and this was the previously best-known estimate for m. We show that the above discretization statement actually holds when m is a universal constant.

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Metric Inequalities

Cited by 13 publications

References 88 publications

Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete

Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete

Heat flow and quantitative differentiation

Nonpositive curvature is not coarsely universal

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