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 $t\geq 1$, $$\max \left{N(f, t \cdot g),\,N(f^*, t \cdot g),\,N(g, t \cdot f),\,N(g, t \cdot f^*)\right} \leq \exp\left( \frac{γ_n^2\, n}{t} \right),$$ where $f^*$ 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 $γ_n \leq c(\ln n)^2$. Our result shows that the isotropic position of a log-concave function already provides an almost $1$-regular functional $M$-position.