Lipschitz extension of multiple Banach-valued functions in the sense of Almgren

Abstract

A multiple-valued function $f:X\to {\mathbf{Q}}_Q(Y)$ 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 $(X,{\mathbf{Q}}_Q(Y))$ has the Lipschitz extension property if $Y$ is a Banach space and $X$ is a metric space with a finite Nagata dimension. We also show that ${\mathbf{Q}}_Q(Y)$ is an absolute Lipschitz retract if $Y$ is a finite algebraic dimensional Banach space.

6 pages

Cited by in corpus (1)

• Kirszbraun's extension theorem fails for Almgren's multiple valued functions