## Journal of the Mathematical Society of Japan

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

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

#### 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.

#### Article information

**Source**

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

**Dates**

Received: 5 June 2017

Revised: 27 December 2017

First available in Project Euclid: 19 March 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1552982427

**Digital Object Identifier**

doi:10.2969/jmsj/78177817

**Mathematical Reviews number (MathSciNet)**

MR3943454

**Zentralblatt MATH identifier**

07090059

**Subjects**

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]

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1552982427