L1 Poincare inequality for an expander graph ($\sum a_{xy}\Vert f(x)-f(y){\Vert }_1\ge \frac{h}{|V|}{\sum }_{x,y}\Vert f(x)-f(y){\Vert }_1$)

If $G$ has adjacency matrix $A=(a_{xy})$ and expansion $h$, then every $L^1$-valued map on its $n$ vertices satisfiesFor scalar functions this follows from the layer-cake formula applied above and below a median; integration over the $L^1$ coordinate proves the vector-valued form.

For a nonnegative measurable function $g$,On a graph, applying this identity to $|g(x)-g(y)|$ expresses its total variation as the integral of the edge boundaries of the superlevel sets.

More generally, if $\phi :[0,\infty )\to [0,\infty )$ is an absolutely continuous function on every compact interval, is increasing, and satisfies $\phi (0)=0$, with its nonnegative derivative defined almost everywhere, then Tonelli theorem and the fundamental theorem of calculus give

Let $F,G\ge 0$, $1<p<\infty$, and $G\in L^p$. Assume the displayed tail inequality and finite positive-threshold superlevel measures. First split $G$ at $a/2$ to obtain $a\mu \{F>2a\}\le {\int }_{\{G>a\}}G$. Layer cake representation and Tonelli theorem imply $F\in L^p$. Integrating the original inequality against $p{a}^{p-2}$ then gives $\Vert F{\Vert }_p^p\le q\int G{F}^{p-1}\le q\Vert G{\Vert }_p\Vert F{\Vert }_p^{p-1}$, with $q=p/(p-1)$. This is the analytic estimate behind the Doob Lp maximal inequality.

Let a connected $d$-regular expander graph have $n$ vertices, edge-expansion constant $h>0$, and $d\ge 2$. For $n\ge 16d^2$, its least bi-Lipschitz distortion into L1 space satisfies the explicit boundNormalize an embedding so that each image distance is at least the graph distance and at most $D$ times it. Every image edge has length at most $D$. The L1 Poincare inequality for an expander graph then gives $D\ge (h/d)\overline{d}$, where $\overline{d}$ is the mean ordered-pair graph distance.

A ball of radius $r$ has at most $1+d+\cdots +d^r\le 2d^r$ vertices. Take $r=\lfloor \log (n/4)/\log d\rfloor$. Its size is at most $n/2$, so at least half the vertices are farther than $r$ from each fixed vertex. Hence $\overline{d}\ge r/2$. The size assumption implies $r\ge (\log n)/(2\log d)$, proving the bound. For a fixed-degree family with expansion bounded below, distortion therefore grows at least proportionally to $\log n$.