The maximum entropy principle and volumetric properties of Orlicz balls

Zakhar Kabluchko; Joscha Prochno

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arXiv · 2007.05247

The maximum entropy principle and volumetric properties of Orlicz balls

Zakhar Kabluchko·Joscha Prochno

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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 Schmuckenschl\"ager [GAFA, Lecture notes in Math. 1469 (1991), 174--178] for $\ell_p^d$-balls. As another application, we determine the precise asymptotic volume ratio for $2$-concave Orlicz spaces $\ell_M^d$. 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 $\ell_p^d$-balls, which are Orlicz balls when the Orlicz function is $M(t) = |t|^p$.

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Zakhar Kabluchko, Joscha Prochno. 2020-07-10. The maximum entropy principle and volumetric properties of Orlicz balls. https://arxiv.org/abs/2007.05247

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