A Generalized Isoperimetric Inequality via Thick Embeddings of Graphs

TL;DR

This paper establishes a generalized isoperimetric inequality using thick embeddings of graphs, demonstrating the existence of low-dilation maps in higher dimensions and providing a counterexample in dimension 3.

Contribution

It introduces a new isoperimetric inequality based on $k$-dilation, linking it to thick embeddings of graphs and extending classical results to higher dimensions.

Findings

Existence of degree 1 maps with bounded $(n-1)$-dilation in dimensions ≥ 4

Counterexample showing such maps cannot exist in dimension 3

Reduction of the inequality to a graph embedding problem using Kolmogorov-Barzdin theorem

Abstract

We prove a generalized isoperimetric inequality for a domain diffeomorphic to a sphere that replaces filling volume with $k$-dilation. Suppose $U$ is an open set in $\mathbb{R}^n$ diffeomorphic to a Euclidean $n$-ball. We show that in dimensions at least 4 there is a map from a standard Euclidean ball of radius about $vol(\partial U)^{1/(n-1)}$ to $U$, with degree 1 on the boundary, and $(n-1)$-dilation bounded by some constant only depending on $n$. We also give an example in dimension 3 of an open set where no such map with small $(n-1)$-dilation can be found. The generalized isoperimetric inequality is reduced to a theorem about thick embeddings of graphs which is proved using the Kolmogorov-Barzdin theorem and the max-flow min-cut theorem. The proof of the counterexample in dimension 3 relies on the coarea inequality and a short winding number computation.