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 $t\geqslant 1$, $$ \begin{align*} &\max\Bigl\{N(f,t\odot g),N(g,t\odot f^{\ast})\Bigr\}\leqslant\exp\!\left(\gamma_{n} n/t\right)\quad\textrm{and}\quad \max\Bigl\{N(f^{\ast},t\odot g),N(g,t\odot f)\Bigr\}\leqslant\exp\!\left(\delta_{n} n/t\right),\end{align*} $$ where $f^{\ast }$ denotes the Legendre dual of $f$, $(t\odot f)(x)=f(x/t)$ is the $t$-homothety of $f$, and $\gamma _{n} \leqslant c(\ln n)^{2}$, $\delta _{n}\leqslant c\ln n$. Our result shows that the isotropic position of a log-concave function already provides an almost $1$-regular functional $M$-position.