Title: The maximum of the 1-measurement of a metric measure space,
Abstract: For a metric measure space, we treat the set of distributions of 1-Lipschitz
functions, which is called the 1-measurement. On the 1-measurement, we have a
partial order relation by the Lipschitz order introduced by Gromov. The aim of
this paper is to study the maximum and maximal elements of the 1-measurement
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.