Limit formulas for metric measure invariants and phase transition property

Ryunosuke Ozawa, Takashi Shioya

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


We generalize the observable diameter and the separation distance for metric measure spaces to those for pyramids, and prove some limit formulas for these invariants for a convergent sequence of pyramids. We obtain various applications of our limit formulas as follows. We have a criterion of the phase transition property for a sequence of metric measure spaces or pyramids, and find some examples of symmetric spaces of noncompact type with the phase transition property. We also give a simple proof of a theorem in Funano and Shioya (Geom Funct Anal 23(3):888–936, 2013) on the limit of an N-Lévy family.

Original languageEnglish
Pages (from-to)759-782
Number of pages24
JournalMathematische Zeitschrift
Issue number3-4
Publication statusPublished - 2015 Aug 26


  • Concentration of measure
  • Dissipation
  • Metric measure space
  • Observable diameter
  • Pyramid
  • Separation distance

ASJC Scopus subject areas

  • Mathematics(all)


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