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 $\ell_p$ 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 $\ell_p$ 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.