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 $σ>0$ such that for any $Δ>0$ there is a distribution over random partitions of $M$ into clusters of diameter at most $Δ$ such that for every $x,y\in M$ the probability that they belong to different clusters is at most $σd_M(x,y)/Δ$. 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 $X$.

Figures (3)

• Figure 1: Given $K\geqslant 1$, the assertion that the Lipschitz extension modulus of a metric space $\mathcal{M}$ satisfies $\mathsf{e}(\mathcal{M})<K$ means that for all subsets $\mathscr{C}\subseteq \mathcal{M}$, all Banach spaces $\mathbf{Z}$ and all$1$-Lipschitz mappings $f:\mathscr{C}\to \mathbf{Z}$, there is a $K$-Lipschitz mapping $F:\mathcal{M}\to \mathbf{Z}$ such that the above diagram commutes, where $\mathsf{Id}_{\mathscr{C}\to \mathcal{M}}:\mathscr{C}\to \mathcal{M}$ is the formal inclusion.
• Figure 2: A schematic depiction of (randomized) iterative ball partitioning of a bounded subset of $\mathbb R^2$, where $\mathbb R^2$ is equipped with a norm whose unit ball is a regular hexagon. The centers of the above hexagons are chosen independently and uniformly at random from a large region that contains the given subset of $\mathbb R^2$. At each step of the iteration, a new hexagon appears, and it carves out a new cluster which consists of the part of the hexagon that does not intersect any of the clusters that have been formed in the previous stages of the iteration. The first few clusters that are formed by this procedure are typically hexagons, but at later stages the clusters become more complicated and less "round." In particular, they can eventually become disconnected, as exhibited by the region that is shaded black above.
• Figure 3: A schematic depiction of the partition of $B_{\mathbf X}$ into the sets $U,V,W$ (with the sets $U,W$ shaded), as well as the line segments parallel to $x$ that are used in the justification of the estimate \ref{['eq:desired shift xt']}.

Theorems & Definitions (228)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4: examples of consequences of Theorem \ref{['thm:sep bounds in overview']}
• Conjecture 6
• Conjecture 7
• Theorem 8: Ball's reverse isoperimetric theorem Bal91-reverse
• Conjecture 9: isomorphic reverse isoperimetry
• Conjecture 10: weak isomorphic reverse isoperimetry
• Conjecture 11: symmetric version of Conjecture \ref{['weak isomorphic reverse conj1']}