Regular functional covering numbers

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 $f:{\mathbb{R}}^n\to [0,\mathrm{\infty })$ satisfies, for all $t\ge 1$, $$\max \{N(f,t\cdot g),N(f^{\ast },t\cdot g),N(g,t\cdot f),N(g,t\cdot f^{\ast })\}\le \exp \left(\frac{{\gamma }_n^{2}n}{t}\right),$$ where $f^{\ast }$ denotes the Legendre dual of $f$, $(t\cdot f)(x)=f(x\mathrm{/}t)$ is the $t$-homothety of $f$, $g(x)=\exp (-\frac{1}{2}\mathrm{\mid }x{\mathrm{\mid }}^2)$ and ${\gamma }_n\le c(\ln n{)}^2$. Our result shows that the isotropic position of a log-concave function already provides an almost $1$-regular functional $M$-position.