Journal of the Mathematical Society of Japan

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


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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.

Article information

J. Math. Soc. Japan, Volume 71, Number 2 (2019), 635-650.

Received: 5 June 2017
Revised: 27 December 2017
First available in Project Euclid: 19 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Secondary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]

metric measure space Lipschitz order 1-measurement isoperimetric inequality observable diameter


NAKAJIMA, Hiroki. The maximum of the 1-measurement of a metric measure space. J. Math. Soc. Japan 71 (2019), no. 2, 635--650. doi:10.2969/jmsj/78177817.

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  • [1] T. Figiel, J. Lindenstrauss and V. D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math., 139 (1977), 53–94.
  • [2] P. Rayón and M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13 (2003), 178–215.
  • [3] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, reprint of the 2001 English edition, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. Based on the 1981 French original; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates.
  • [4] P. Lévy, Problèmes concrets d'analyse fonctionnelle, Avec un complément sur les fonctionnelles analytiques par F. Pellegrino, Gauthier-Villars, Paris, 1951 (French).
  • [5] T. Shioya, Metric measure geometry, Gromov's theory of convergence and concentration of metrics and measures, IRMA Lect. Math. Theor. Phys., 25, EMS Publishing House, Zürich, 2016.
  • [6] C. T. Yang, Odd-dimensional wiedersehen manifolds are spheres, J. Differential Geom., 15 (1980), 91–96.