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 $(Z,|\cdot|Z)$ 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 $\alpha$-H\"older with constant $1$, yet the $\alpha$-H\"older constant of any $F:X\to \ell_2$ that extends $f$ satisfies $$ |F|{\mathrm{Lip}(\alpha)}\gtrsim (\log n)^{\frac{2\alpha-1}{4\alpha}}+\left(\frac{\log n}{\log\log n}\right)^{\alpha^2-\frac12}. $$ 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 K\"onig, 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.