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**The maximum of the 1-measurement of a metric measure space.**
*(English)*
Zbl 1420.53051

Summary: 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 [Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. 3rd printing. Basel: Birkhäuser (2007; Zbl 1113.53001)]. 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.

### MSC:

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

53C20 | Global Riemannian geometry, including pinching |

### Keywords:

metric measure space; Lipschitz order; 1-measurement; isoperimetric inequality; observable diameter### Citations:

Zbl 1113.53001### References:

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[2] | P. Rayón and M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13 (2003), 178-215. · Zbl 1044.46057 · doi:10.1007/s000390300002 |

[3] | 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. · Zbl 1335.53003 |

[4] | C. T. Yang, Odd-dimensional wiedersehen manifolds are spheres, J. Differential Geom., 15 (1980), 91-96. · Zbl 0491.53039 · doi:10.4310/jdg/1214435386 |

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