Abstract
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 language | English |
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Pages (from-to) | 759-782 |
Number of pages | 24 |
Journal | Mathematische Zeitschrift |
Volume | 280 |
Issue number | 3-4 |
DOIs | |
Publication status | Published - 2015 Aug 26 |
Keywords
- Concentration of measure
- Dissipation
- Metric measure space
- Observable diameter
- Pyramid
- Separation distance
ASJC Scopus subject areas
- Mathematics(all)