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

A Generalized Isoperimetric Inequality via Thick Embeddings of Graphs