Exponential profiles for the intrinsic volumes of Orlicz balls
Abstract
An Orlicz ball is the set of points where the sum of a fixed even convex function of the coordinates does not exceed a given radius, here proportional to the dimension. For a broad class of such functions, including non-homogeneous ones, we find the exponential rate of the intrinsic volumes whose order is proportional to the dimension and express it by a two-parameter variational formula involving only one-dimensional integrals. As a function of the order-to-dimension ratio, this rate is concave and continuous on the closed unit interval, and the convergence is uniform in the order. This proves the exponential-profile conjecture of Kabluchko, Marynych, and Prochno for these bodies and recovers their result for ℓp balls with p>1. The proof relies on the theory of large deviations. An exact formula for the Laplace transform of the intrinsic volumes in the radius, extending theirs for ℓp balls, plays the role of a moment generating function, and a Laplace method gives its exponential rate. An exponential change of measure, analogous to that used in Cramér's theorem, then recovers the intrinsic volumes. The key step is a lower bound on the tilted density at its mean, obtained from log-concavity rather than a law of large numbers.
Related
- Sudakov minoration for weighted Orlicz balls
- BMO-Type characterizations and endpoint estimates for commutators of intrinsic square functions on Orlicz-Hardy spaces
- Non-centered maximal operators over ℓq balls for exponential and Laguerre-type measures
- Mellin transformed exponential volumes and probabilistic moduli spaces
- Ornstein--Uhlenbeck spirals: Absorption Probabilities and Conic Intrinsic Volumes
BibTeX
@misc{sadok2026exponential,
title = {{Exponential profiles for the intrinsic volumes of Orlicz balls}},
author = {Mykyta Sadok},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16904},
eprint = {2610.16904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16904},
}
RIS
TY - UNPB
TI - Exponential profiles for the intrinsic volumes of Orlicz balls
AU - Mykyta Sadok
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16904
UR - https://arxiv.org/abs/2610.16904
AB - An Orlicz ball is the set of points where the sum of a fixed even convex function of the coordinates does not exceed a given radius, here proportional to the dimension. For a broad class of such functions, including non-homogeneous ones, we find the exponential rate of the intrinsic volumes whose order is proportional to the dimension and express it by a two-parameter variational formula involving only one-dimensional integrals. As a function of the order-to-dimension ratio, this rate is concave and continuous on the closed unit interval, and the convergence is uniform in the order. This proves the exponential-profile conjecture of Kabluchko, Marynych, and Prochno for these bodies and recovers their result for ℓ_p balls with p>1. The proof relies on the theory of large deviations. An exact formula for the Laplace transform of the intrinsic volumes in the radius, extending theirs for ℓ_p balls, plays the role of a moment generating function, and a Laplace method gives its exponential rate. An exponential change of measure, analogous to that used in Cramér's theorem, then recovers the intrinsic volumes. The key step is a lower bound on the tilted density at its mean, obtained from log-concavity rather than a law of large numbers.
ER -