Regular functional covering numbers
Apostolos Giannopoulos, Natalia Tziotziou
TL;DR
The paper extends Pisier's regular M-position framework to the functional setting of geometric log-concave functions, proving that such functions admit a regular functional M-position and that isotropic positions yield near-optimal, scale-invariant covering bounds. It establishes explicit exponential-type bounds on functional covering numbers involving the Gaussian benchmark, with constants γ_n and δ_n that scale polylogarithmically with dimension. Duality plays a central role: the Legendre dual f* and the polarity-based f_A also satisfy corresponding regular covering estimates, linking functional results to classical convex-geometry inequalities. These findings provide a robust functional analogue of Milman’s M-positions, enabling uniform control of covering behavior for log-concave functions and potential applications in high-dimensional analysis of such functions and measures.
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.