The main result is that a Banach space X is not super-reflexive if and only if the diamond graphs Dn Lipschitz embed into X with distortions independent of n. One of the consequences of that and previously known results is that dimension reduction a-la Johnson–Lindenstrauss fails in any non super reflexive space with non trivial type. We also introduce the concept of Lipschitz (p, r)-summing ma...

We examine the relationship between infinity harmonic functions, absolutely minimizing Lipschitz extensions, strong absolutely minimizing Lipschitz extensions, and absolutely gradient minimizing extensions in CarnotCarathéodory spaces. Using the weak Fubini property we show that absolutely minimizing Lipschitz extensions are infinity harmonic in any sub-Riemannian manifold.

Remark 1. The Lipschitz condition (4) is called global because it holds for all x, y ∈ R with the same constant L. Klebaner Theorem 5.4 assumes only a local Lipschitz condition, where the Lipschitz constant may depend on the size of x, y. We use a global condition in order to make the presentation simpler. Later on we will also assume a Lipschitz condition with respect to t, see (21). (There is...

This paper treats near-rings of zero-preserving Lipschitz functions on metric spaces that are also abelian groups, using pointwise addition of functions as addition and composition of functions as multiplication. We identify a condition on the metric ensuring that the set of all such Lipschitz functions is a near-ring, and we investigate the complications that arise from the lack of left distri...

Abstract. 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...

A multiple-valued function f : X → QQ(Y ) is essentially a rule assigning Q unordered and non necessarily distinct elements of Y to each element of X. We study the Lipschitz extension problem in this context by using two general Lipschitz extension theorems recently proved by U. Lang and T. Schlichenmaier. We prove that the pair ` X,QQ(Y ) ́ has the Lipschitz extension property if Y is a Banach ...

We prove the existence of a (random) Lipschitz function F : Z → Z such that, for every x ∈ Z, the site (x, F (x)) is open in a site percolation process on Z. The Lipschitz constant may be taken to be 1 when the parameter p of the percolation model is sufficiently close to 1.

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where “almost everywhere” refers to the Lebesgue measure. Our main result is an extension of this theorem where the Lebesgue measure is replaced by an arbitrary measure μ. In particular we show that the differentiability properties of Lipschitz functions at μ-almost every point ar...