We study the Prohorov distance between the distance matrix distributions of two metric measure spaces. We prove that it is not smaller than 1-box distance between two metric measure spaces and also prove that it is not larger than 0-box distance between two metric measure spaces.
"Distance between metric measure spaces and distance matrix distributions." Tsukuba J. Math. 38 (2) 159 - 170, March 2015. https://doi.org/10.21099/tkbjm/1429103718