Regular functional covering numbers
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912936463695872 |
|---|---|
| author | Giannopoulos, Apostolos Tziotziou, Natalia |
| author_facet | Giannopoulos, Apostolos Tziotziou, Natalia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04301 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regular functional covering numbers Giannopoulos, Apostolos Tziotziou, Natalia Metric Geometry Primary 52A23, Secondary 46B06, 52A40, 52C17, 26B25 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. |
| title | Regular functional covering numbers |
| topic | Metric Geometry Primary 52A23, Secondary 46B06, 52A40, 52C17, 26B25 |
| url | https://arxiv.org/abs/2512.04301 |