A Generalized Isoperimetric Inequality via Thick Embeddings of Graphs

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.