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

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

2006-09-21

arxiv.org/abs/math/0609606v1

Mathematics

Metric Geometry

6 pages

Scientific paper

A multiple-valued function $f:X\to {\bf Q}_Q(Y)$ 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 $(X,{\bf Q}_Q(Y))$ 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 ${\bf Q}_{Q}(Y)$ is an absolute Lipschitz retract if $Y$ is a finite algebraic dimensional Banach space.

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