Undistorted fillings in subsets of metric spaces

TL;DR

This paper demonstrates that in metric spaces with finite Nagata dimension, certain connectivity and isoperimetric properties imply isoperimetric undistortion up to a specific dimension, with broad implications for geometric analysis.

Contribution

It generalizes previous results by establishing isoperimetric undistortion under broader conditions in metric spaces with finite Nagata dimension.

Findings

Lipschitz k-connectedness implies Euclidean isoperimetric inequalities

Euclidean isoperimetric inequalities imply coning inequalities

Proves an analog of Federer-Fleming deformation theorem in these spaces

Abstract

We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite Nagata dimension and is Lipschitz $k$-connected or admits Euclidean isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is isoperimetrically undistorted in $Y$ up to dimension $k+1$. This generalizes and strengthens a recent result of the third named author and has several consequences and applications. It yields for example that in spaces of finite Nagata dimension, Lipschitz connectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities. It furthermore allows us to prove an analog of the Federer-Fleming deformation theorem in spaces of finite Nagata dimension admitting Euclidean isoperimetric inequalities.