TY - JOUR

T1 - Undistorted fillings in subsets of metric spaces

AU - Basso, Giuliano

AU - Wenger, Stefan

AU - Young, Robert

N1 - Funding Information:
G. B. and S. W. were supported by Swiss National Science Foundation grant 182423 . R. Y. was supported by National Science Foundation grant 2005609 .
Publisher Copyright:
© 2023 The Authors

PY - 2023/6/15

Y1 - 2023/6/15

N2 - 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.

AB - 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.

KW - Coning inequalities

KW - Deformation theorem

KW - Integral currents

KW - Isoperimetric inequalities

KW - Isoperimetric subspace distortion

KW - Lipschitz connectivity

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U2 - 10.1016/j.aim.2023.109024

DO - 10.1016/j.aim.2023.109024

M3 - Article

AN - SCOPUS:85152716252

SN - 0001-8708

VL - 423

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 109024

ER -