Undistorted fillings in subsets of metric spaces

Lipschitz k -connectivity, Euclidean isoperimetric inequalities, and coning inequalities all measure the difficulty of filling a k -dimensional cycle in a space by a (k+1)-dimensional object. In many cases, such as Banach spaces and CAT(0) spaces, it is easy to prove Lipschitz connectivity or a coning inequality, but harder to obtain a Euclidean isoperimetric inequality. We show that in spaces of finite Nagata dimension, Lipschitz con-nectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequali-ties. We show this by proving that if X has finite Nagata dimension and is Lipschitz k -connected or admits Euclidean isoperimetric inequalities up to dimension k then any isomet-ric embedding of X into a metric space is isoperimetrically undistorted up to dimension k + 1. Since X embeds in L infinity, which admits a Euclidean isoperimetric inequality and a con-ing inequality, X admits such inequalities as well. In addition, we prove that an analog of the Federer-Fleming deformation theorem holds in such spaces X and use it to show that if X has finite Nagata dimension and is Lipschitz k -connected, then integral (k + 1)-currents in X can be approximated by Lipschitz chains in total mass. (c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

Automated summary

This article investigates Lipschitz k-connectivity, Euclidean isoperimetric inequalities, and coning inequalities. It shows that Lipschitz connectivity implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities, in spaces with finite Nagata dimension. Additionally, it proves that Lipschitz k-connectivity in such spaces can be approximated by Lipschitz chains in total mass.