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Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

100%

Mendel M., Naor A.

Analysis and Geometry in Metric Spaces

2013

tom 1

163-199

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 bi-Lipschitz 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.

Comparison of Metric Spectral Gaps

100%

Naor A.

Analysis and Geometry in Metric Spaces

2014

tom 2

nr 1

Let A = (aij) ∊ Mn(ℝ) be an n by n symmetric stochastic matrix. For p ∊ [1, ∞) and a metric space (X, dX), let γ(A, dpx) be the infimum over those γ ∊ (0,∞] for which every x1, . . . , xn ∊ X satisfy [...] Thus γ (A, dpx) measures the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel dpX : X × X →[0,∞). We study pairs of metric spaces (X, dX) and (Y, dY ) for which there exists Ψ: (0,∞)→(0,∞) such that γ (A, dpX) ≤Ψ (A, dpY ) for every symmetric stochastic A ∊ Mn(ℝ) with (A, dpY ) < ∞. When Ψ is linear a complete geometric characterization is obtained. Our estimates on nonlinear spectral gaps yield new embeddability results as well as new nonembeddability results. For example, it is shown that if n ∊ ℕ and p ∊ (2,∞) then for every f1, . . . , fn ∊ Lp there exist x1, . . . , xn ∊ L2 such that [...] and [...] This statement is impossible for p ∊ [1, 2), and the asymptotic dependence on p in (0.1) is sharp. We also obtain the best known lower bound on the Lp distortion of Ramanujan graphs, improving over the work of Matoušek. Links to Bourgain-Milman-Wolfson type and a conjectural nonlinear Maurey-Pisier theorem are studied.

Ograniczanie wyników

2 Analysis and Geometry in Metric Spaces

2 Naor A.

1 Mendel M.

1 2014

1 2013

Wyniki wyszukiwania

Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

Comparison of Metric Spectral Gaps