Extension, separation and isomorphic reverse isoperimetry

Abstract

The Lipschitz extension modulus $e(M)$ of a metric space $M$ is the infimum over $L\ge 1$ such that for any Banach space $Z$ and any $C\subset M$, any 1-Lipschitz function $f:C\to Z$ can be extended to an $L$-Lipschitz function $F:M\to Z$. Johnson, Lindenstrauss and Schechtman proved that if $X$ is an $n$-dimensional normed space, then $e(X)=O(n)$. In the reverse direction, we prove that every $n$-dimensional normed space $X$ satisfies $e(X)\ge n^c$, where $c>0$ is a universal constant. Our core technical contribution is a geometric structural result on stochastic clustering of finite dimensional normed spaces which implies upper bounds on their Lipschitz extension moduli using an extension method of Lee and the author. The separation modulus of a metric space $(M,d_M)$ is the infimum over $\sigma>0$ such that for any $\Delta>0$ there is a distribution over random partitions of $M$ into clusters of diameter at most $\Delta$ such that for every $x,y\in M$ the probability that they belong to different clusters is at most $\sigma d_M(x,y)/\Delta$. We obtain upper and lower bounds on the separation moduli of finite dimensional normed spaces that relate them to well-studied volumetric invariants. Using these connections, we find the growth rate of the separation moduli of various normed spaces. We formulate a conjecture on isomorphic reverse isoperimetry that can be used with our volumetric bounds on the separation modulus to obtain many more asymptotic evaluations of the separation moduli of normed spaces. Our estimates on the separation modulus imply improved bounds on the Lipschitz extension moduli of various classical spaces. In particular, we deduce an improved bound on $e(\ell_p^n)$ when $p>2$ that resolves a conjecture of Brudnyi and Brudnyi, and prove that $e(\ell_\infty^n)\asymp{\sqrt{n}}$, which is the first time that the order of $e(X)$ has been evaluated for any normed space $X$.