On Lipschitz extension from finite subsets

Abstract

We prove that for every $n\in \mathbb{N}$ there exists a metric space $(X,d_X)$, an $n$-point subset $S\subseteq X$, a Banach space $(Z,\Vert \cdot {\Vert }_Z)$ and a $1$-Lipschitz function $f:S\to Z$ such that the Lipschitz constant of every function $F:X\to Z$ that extends $f$ is at least a constant multiple of $\sqrt{\log n}$. This improves a bound of Johnson and Lindenstrauss. We also obtain the following quantitative counterpart to a classical extension theorem of Minty. For every $\alpha \in (1/2,1]$ and $n\in \mathbb{N}$ there exists a metric space $(X,d_X)$, an $n$-point subset $S\subseteq X$ and a function $f:S\to {\ell }_2$ that is $\alpha$-Hölder with constant $1$, yet the $\alpha$-Hölder constant of any $F:X\to {\ell }_2$ that extends $f$ satisfies $$\Vert F{\Vert }_{\mathrm{Lip}(\alpha )}\gtrsim (\log n{)}^{\frac{2\alpha -1}{4\alpha }}+{\left(\frac{\log n}{\log \log n}\right)}^{{\alpha }^2-\frac{1}{2}}.$$ We formulate a conjecture whose positive solution would strengthen Ball's nonlinear Maurey extension theorem, serving as a far-reaching nonlinear version of a theorem of König, Retherford and Tomczak-Jaegermann. We explain how this conjecture would imply as special cases answers to longstanding open questions of Johnson and Lindenstrauss and Kalton.