Regular functional covering numbers

Abstract

We establish the existence of a regular functional $M$-position, in the sense of Pisier, for geometric log-concave functions. This provides a functional analogue of Pisier's regular $M$-positions for convex bodies and yields uniform control of covering numbers at all scales. Specifically, we show that every isotropic geometric log-concave function $f:{\mathbb{R}}^n\to [0,\infty )$ satisfies, for all $t\ge 1$, $$\max \left\{N(f, t \cdot g),\,N(f^*, t \cdot g),\,N(g, t \cdot f),\,N(g, t \cdot f^*)\right\} \leq \exp\left( \frac{γ_n^2\, n}{t} \right),$$ where $f^{\ast }$ denotes the Legendre dual of $f$, $(t\cdot f)(x)=f(x/t)$ is the $t$-homothety of $f$, $g(x)=\exp \left(-\frac{1}{2}|x{|}^2\right)$ and ${\hat{I}}_n^3\le c(\ln n{)}^2$. Our result shows that the isotropic position of a log-concave function already provides an almost $1$-regular functional $M$-position.

18 pages, International Mathematics Research Notices (to appear)