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 $k=γn$, where $γ\in (1/\sqrt{n},1)$, satisfies $K\cap F\subseteq \frac{c}{γ}\sqrt{n}L_K (B_2^n\cap F)$ with probability greater than $1-\exp (-\sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(μ)$ of an isotropic log-concave probability measure $μ$ on ${\mathbb R}^n$. For every $1\leq q\leq n$ and $γ\in (0,1)$ we show that a random subspace $F\in G_{n,(1-γ)n}$ satisfies $Z_q(μ)\cap F\subseteq c_2(γ)\sqrt{q}\,B_2^n\cap F$. We also give bounds on the diameter of random projections of $Z_q(μ)$ 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 $O(\log^2n)$.

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