Definition 1. Let X and Y be metric spaces. Let f: X → Y. If
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چکیده
In this lecture we will discuss several results on Lipschitz extensions of mappings. In particular we’ll prove the Kirszbraun extension theorem for mappings of subsets of `2 to ` m 2 . The book [4] gives a complete account of the proof and many other related results. We’ll also present a result by Johnson and Lindenstrauss [2] on extensions of mappings from finite subsets of arbitrary metric spaces into `2 . Finally, we will present a nice application of the Kirszbraun extension theorem in showing that embedding spheres into the plane requires Ω( √ n) distortion. We also show that this bound is tight.
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A space $Y$ is called an {em extension} of a space $X$, if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {em equivalent}, if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Yleq Y'$, if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ ...
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