Geometry of random sections of isotropic convex bodies

Abstract

Let $K$ be an isotropic symmetric convex body in ${\mathbb{R}}^n$. We show that a subspace $F\in G_{n,n-k}$ of codimension $k=\gamma n$, where $\gamma \in (1/\sqrt{n},1)$, satisfies $$K\cap F\subseteq \frac{c}{\gamma }\sqrt{n}{L}_K(B_2^n\cap F)$$ with probability greater than $1-\exp (-\sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(\mu )$ of an isotropic log-concave probability measure $\mu$ on ${\mathbb{R}}^n$. For every $1\le q\le n$ and $\gamma \in (0,1)$ we show that a random subspace $F\in G_{n,(1-\gamma )n}$ satisfies $Z_q(\mu )\cap F\subseteq c_2(\gamma )\sqrt{q}{B}_2^n\cap F$. We also give bounds on the diameter of random projections of $Z_q(\mu )$ and using them we deduce that if $K$ is an isotropic convex body in ${\mathbb{R}}^n$ then for a random subspace $F$ of dimension $(\log n{)}^4$ one has that all directions in $F$ are sub-Gaussian with constant $O({\log }^{2}n)$.

arXiv commentTo appear in the Bulletin of the Hellenic Mathematical Society