2022/11/17 by Portnoy, Elia
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2211.09900
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 ℝn 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(∂ 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.