The maximum of the 1-measurement of a metric measure space

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Abstract

For a metric measure space, we consider the set of distributions of 1-Lipschitz functions, which is called the 1-measurement. On the 1-measurement, we have the Lipschitz order relation introduced by M. Gromov. The aim of this paper is to study the maximum and maximal elements of the 1-measurement of a metric measure space with respect to the Lipschitz order. We present a necessary condition of a metric measure space for the existence of the maximum of the 1-measurement. We also consider a metric measure space that has the maximum of its 1-measurement.

Original languageEnglish
Pages (from-to)635-650
Number of pages16
JournalJournal of the Mathematical Society of Japan
Volume71
Issue number2
DOIs
Publication statusPublished - 2019 Jan 1

Keywords

  • 1-measurement
  • Isoperimetric inequality
  • Lipschitz order
  • Metric measure space
  • Observable diameter

ASJC Scopus subject areas

  • Mathematics(all)

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