Undistorted fillings in subsets of metric spaces

Abstract

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 connectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities. 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 isometric embedding of X into a metric space is isoperimetrically undistorted up to dimension k+1. Since X embeds in L∞, which admits a Euclidean isoperimetric inequality and a coning 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.