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 $Fin G_{n,n-k}$ of codimension $k=gamma n$, where $gammain (1/sqrt{n},1)$, satisfies $$Kcap Fsubseteq frac{c}{gamma }sqrt{n}L_K (B_2^ncap F)$$ 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 $1leq qleq n$ and $gammain (0,1)$ we show that a random subspace $Fin G_{n,(1-gamma )n}$ satisfies $Z_q(mu )cap Fsubseteq c_2(gamma )sqrt{q},B_2^ncap F$. 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 $O(log^2n)$.