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15 February 2022 Filling metric spaces
Yevgeny Liokumovich, Boris Lishak, Alexander Nabutovsky, Regina Rotman
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Duke Math. J. 171(3): 595-632 (15 February 2022). DOI: 10.1215/00127094-2021-0039

Abstract

We prove a new version of isoperimetric inequality: Given a positive real m, a Banach space B, a closed subset Y of metric space X, and a continuous map f:YB with f(Y) compact

infFHCm+1(F(X))c(m)HCm(f(Y))m+1m,

where HCm denotes the m-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps F:XB such that F(y)=f(y) for all yY, and c(m) depends only on m. Moreover, one can find F with a nearly minimal HCm+1 such that its image lies in the C(m)HCm(f(Y))1m-neighborhood of f(Y) with the exception of a subset with zero (m+1)-dimensional Hausdorff measure.

The paper also contains a very general coarea inequality for Hausdorff content and its modifications.

As an application we demonstrate an inequality conjectured by Larry Guth that relates the m-dimensional Hausdorff content of a compact metric space with its (m1)-dimensional Urysohn width. We show that this result implies new systolic inequalities that both strengthen the classical Gromov’s systolic inequality for essential Riemannian manifolds and extend this inequality to a wider class of non-simply-connected manifolds.

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Yevgeny Liokumovich. Boris Lishak. Alexander Nabutovsky. Regina Rotman. "Filling metric spaces." Duke Math. J. 171 (3) 595 - 632, 15 February 2022. https://doi.org/10.1215/00127094-2021-0039

Information

Received: 25 September 2019; Revised: 31 January 2021; Published: 15 February 2022
First available in Project Euclid: 17 February 2022

Digital Object Identifier: 10.1215/00127094-2021-0039

Subjects:
Primary: 28A78
Secondary: 49Q15 , 51F99 , 53C23

Keywords: Hausdorff content , Isoperimetric inequality , macroscopic scalar curvature , systolic inequality , Urysohn width

Rights: Copyright © 2022 Duke University Press

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Vol.171 • No. 3 • 15 February 2022
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