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Multidimensional geometry, functions of very many variables, and probability. (English) Zbl 1384.60045

Theory Probab. Appl. 59, No. 3, 481-493 (2015); translation from Teor. Veroyatn. Primen. 59, No. 3, 436–451 (2015).
Summary: The diverse facts (from statistical physics, probability theory, dynamical systems, geometry, analysis) which feature in this paper are known to the experts, but as independent, isolated observations. What we do here is interpret and discuss them as instants of the same phenomenon of measure concentration, which occurs in multidimensional spaces and holds for functions of very many variables. Such functions show a tendency towards being almost constant from the point of view of the observer. We also prove a general theorem on the concentration of the volume in a multidimensional domain.

MSC:

60D05 Geometric probability and stochastic geometry
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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