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

Hiroki Nakajima

doi:10.1007/s12220-021-00773-3

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

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we obtain the stability of isoperimetric inequalities with respect to the concentrate topology. The concentration topology is weaker than the □ -topology which is like the weak topology. As an application, we obtain isoperimetric inequalities on the non-discrete n-dimensional l¹-cube and l¹-torus by taking the limits of isoperimetric inequalities of discrete l¹-cubes and l¹-torus. The method of this paper builds on by introducing an ε-relaxed (iso-)Lipschitz order.

Original language	English
Article number	35
Journal	Journal of Geometric Analysis
Volume	32
Issue number	1
DOIs	https://doi.org/10.1007/s12220-021-00773-3
Publication status	Published - 2022 Jan
Externally published	Yes

Keywords

1-measurement
Concentration topology
Isoperimetric inequality
l-Minkowski
Lipschitz order
Metric measure space
Observable diameter
Observable distance
The concentration of measure phenomenon
Torus with l-Minkowski metric

ASJC Scopus subject areas

Geometry and Topology

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10.1007/s12220-021-00773-3

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title = "Isoperimetric Inequality on a Metric Measure Space and Lipschitz Order with an Additive Error",

abstract = "In this paper, we obtain the stability of isoperimetric inequalities with respect to the concentrate topology. The concentration topology is weaker than the □ -topology which is like the weak topology. As an application, we obtain isoperimetric inequalities on the non-discrete n-dimensional l1-cube and l1-torus by taking the limits of isoperimetric inequalities of discrete l1-cubes and l1-torus. The method of this paper builds on by introducing an ε-relaxed (iso-)Lipschitz order.",

keywords = "1-measurement, Concentration topology, Isoperimetric inequality, l-Minkowski, Lipschitz order, Metric measure space, Observable diameter, Observable distance, The concentration of measure phenomenon, Torus with l-Minkowski metric",

author = "Hiroki Nakajima",

note = "Funding Information: The author was supported by JSPS KAKENHI Grant Number 19J10866. Publisher Copyright: {\textcopyright} 2021, Mathematica Josephina, Inc.",

year = "2022",

month = jan,

doi = "10.1007/s12220-021-00773-3",

language = "English",

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journal = "Journal of Geometric Analysis",

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T1 - Isoperimetric Inequality on a Metric Measure Space and Lipschitz Order with an Additive Error

AU - Nakajima, Hiroki

PY - 2022/1

Y1 - 2022/1

N2 - In this paper, we obtain the stability of isoperimetric inequalities with respect to the concentrate topology. The concentration topology is weaker than the □ -topology which is like the weak topology. As an application, we obtain isoperimetric inequalities on the non-discrete n-dimensional l1-cube and l1-torus by taking the limits of isoperimetric inequalities of discrete l1-cubes and l1-torus. The method of this paper builds on by introducing an ε-relaxed (iso-)Lipschitz order.

AB - In this paper, we obtain the stability of isoperimetric inequalities with respect to the concentrate topology. The concentration topology is weaker than the □ -topology which is like the weak topology. As an application, we obtain isoperimetric inequalities on the non-discrete n-dimensional l1-cube and l1-torus by taking the limits of isoperimetric inequalities of discrete l1-cubes and l1-torus. The method of this paper builds on by introducing an ε-relaxed (iso-)Lipschitz order.

KW - 1-measurement

KW - Concentration topology

KW - Isoperimetric inequality

KW - l-Minkowski

KW - Lipschitz order

KW - Metric measure space

KW - Observable diameter

KW - Observable distance

KW - The concentration of measure phenomenon

KW - Torus with l-Minkowski metric

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DO - 10.1007/s12220-021-00773-3

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JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

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ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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