Geometry of random sections of isotropic convex bodies

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 $1\leq q\leq n$ 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 $O(\log^2n)$.