Discrete Rearrangements and the Polya-Szego Inequality on Graphs

TL;DR

This paper extends the Polya-Szeg ext{"o}} inequality to discrete graphs by establishing conditions under which rearrangements on graphs like grids and trees satisfy similar inequalities, bridging continuous and discrete analysis.

Contribution

The paper introduces conditions on graphs that ensure rearrangements satisfy the Polya-Szeg ext{"o}} inequality, including for grid graphs and infinite regular trees, in discrete settings.

Findings

Rearrangements on grid graphs satisfy the inequality in $L^1$.

Optimal ordering in vertex-isoperimetry implies the $L^{ ext{infinity}}$ case.

Canonical rearrangement on infinite regular trees satisfies the inequality for all $p$.

Abstract

For any $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ the symmetric decreasing rearrangement $f^*$ satisfies the Polya-Szeg\H{o} inequality $\| \nabla f^*\|_{L^p} \leq \| \nabla f\|_{L^p}$. The goal of this paper is to establish analogous results in the discrete setting for graphs satisfying suitable conditions. We prove that if the edge-isoperimetric problem on a graph has a sequence of nested minimizers, then this sequence gives rise to a rearrangement satisfying the Polya-Szeg\H{o} inequality in $L^1$. This shows, for example, that a specific rearrangement on the grid graph $\mathbb{Z}^2$, going around the origin in a spiral-like manner, satisfies $\| \nabla f^*\|_{L^1} \leq \| \nabla f\|_{L^1}$. The $L^{\infty}-$case is implied by an optimal ordering condition in vertex-isoperimetry. We use these ideas to prove that the canonical rearrangement on the infinite $d-$regular tree satisfies the Polya-Szeg\H{o} inequality for all $1 \leq p \leq \infty$.