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Isoperimetric Inequality on a Metric Measure Space and Lipschitz Order with an Additive Error

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Abstract

In this paper, we obtain the stability of isoperimetric inequalities with respect to the concentrate topology. The concentration topology is weaker than the \(\square \)-topology which is like the weak topology. As an application, we obtain isoperimetric inequalities on the non-discrete n-dimensional \(l^1\)-cube and \(l^1\)-torus by taking the limits of isoperimetric inequalities of discrete \(l^1\)-cubes and \(l^1\)-torus. The method of this paper builds on by introducing an \(\varepsilon \)-relaxed (iso-)Lipschitz order.

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Acknowledgements

The author would like to thank Professor Takashi Shioya for many helpful suggestions. He also thanks Dr. Daisuke Kazukawa and Dr. Shinichiro Kobayashi for many stimulating discussions.

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  1. Tohoku Daigaku, Sendai, Japan

    Hiroki Nakajima

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  1. Hiroki Nakajima
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Correspondence to Hiroki Nakajima.

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The author was supported by JSPS KAKENHI Grant Number 19J10866.

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Nakajima, H. Isoperimetric Inequality on a Metric Measure Space and Lipschitz Order with an Additive Error. J Geom Anal 32, 35 (2022). https://doi.org/10.1007/s12220-021-00773-3

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