Kirszbraun’s Theorem states that every Lipschitz map S → Rn, where S ⊆ Rm, has an extension to a Lipschitz map Rm → Rn with the same Lipschitz constant. Its proof relies on Helly’s Theorem: every family of compact subsets of Rn, having the property that each of its subfamilies consisting of at most n + 1 sets share a common point, has a non-empty intersection. We prove versions of these theorem...

In this note we show that reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”) can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α : H → Rm is injective, with (α(x))k = |〈x, fk〉|, where {f1, · · · , fm} is a frame for the Hilbert space H, then there exists a left inverse map ω : Rm → H that is Li...

The implicit function theorem asserts that there exists a ball of nonzero radius within which one can express a certain subset of variables, in a system of equations, as functions of the remaining variables. We derive a lower bound for the radius of this ball in the case of Lipschitz maps. Under a sign-preserving condition, we prove that an implicit function exists in the case of a set of inequ...

This is one of a series of papers on Lipschitz maps from metric spaces to L. Here we present the details of results which were announced in [CK06, Section 1.8]: a new approach to the infinitesimal structure of Lipschitz maps into L, and, as a first application, an alternative proof of the main theorem of [CK06], that the Heisenberg group does not admit a bi-Lipschitz embedding in L. The proof u...

Abstract Let $\{X_n\}_{n\in{\mathbb{N}}}$ be an ${\mathbb{X}}$ -valued iterated function system (IFS) of Lipschitz maps defined as $X_0 \in {\mathbb{X}}$ and for $n\geq 1$ , $X_n\;:\!=\;F(X_{n-1},\vartheta_n)$ where $\{\vartheta_n\}_{n \ge 1}$ are independent identically distributed random variables with common probability distribution $\mathfrak{p}$ $F(\cdot,\cdot)$ is continuous in the first ...