Geometry of the $L_q$-centroid bodies of an isotropic log-concave measure

TL;DR

This paper investigates the geometric properties of $L_q$-centroid bodies of isotropic log-concave measures, providing bounds on projections, covering numbers, and regularity, advancing understanding of their structure in high-dimensional convex geometry.

Contribution

It offers new bounds on the inradius of projections, estimates covering numbers, and establishes regularity properties of $Z_q(rac{ ext{body}}{ ext{body}})$, a significant step in high-dimensional convex analysis.

Findings

Bounds on the inradius of random projections of $Z_q(rac{ ext{body}}{ ext{body}})$

Estimates for covering numbers $N( ext{scaled } B_2^n, t Z_q(rac{ ext{body}}{ ext{body}}))$

Proof that $Z_q(rac{ ext{body}}{ ext{body}})$ is a $eta$-regular convex body

Abstract

We study some geometric properties of the $L_q$-centroid bodies $Z_q(\mu)$ of an isotropic log-concave measure $\mu $ on ${\mathbb R}^n$. For any $2\ls q\ls\sqrt{n}$ and for $\varepsilon \in (\varepsilon_0(q,n),1)$ we determine the inradius of a random $(1-\varepsilon)n$-dimensional projection of $Z_q(\mu)$ up to a constant depending polynomially on $\varepsilon $. Using this fact we obtain estimates for the covering numbers $N(\sqrt{\smash[b]{q}}B_2^n,tZ_q(\mu))$, $t\gr 1$, thus showing that $Z_q(\mu)$ is a $\beta $-regular convex body. As a consequence, we also get an upper bound for $M(Z_q(\mu))$.