The maximum entropy principle and volumetric properties of Orlicz balls

Abstract We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a result of Schechtman and Schmuckenschlager (1991) [32] for l p d -balls. As another application, we determine the precise asymptotic volume ratio for 2-concave Orlicz spaces l M d . Our method rests on ideas from statistical mechanics and large deviations theory, more precisely the maximum entropy or Gibbs principle for non-interacting particles, and presents a natural approach and fresh perspective to such geometric and volumetric questions. In particular, our approach explains how the p-generalized Gaussian distribution occurs in problems related to the geometry of l p d -balls, which are Orlicz balls when the Orlicz function is M ( t ) = | t | p .

• Publication year

  2020

• Venue

  Journal of Mathematical Analysis and Applications

• Publication date

  2020-07-10

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1016/j.jmaa.2020.124687arXiv 2007.05247
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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