High dimensional random sections of isotropic convex bodies

We study two properties of random high dimensional sections of convex bodies. In the first part of the paper we estimate the central section function | K ∩ F ⊥ | n − k 1 / k for random F ∈ G n , k and K ⊂ R n a centrally symmetric isotropic convex body. This partially answers a question raised by V.D. Milman and A. Pajor (see [V.D. Milman, A. Pajor, Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Lecture Notes in Math., vol. 1376, Springer, 1989, p. 88]). In the second part we show that every symmetric convex body has random high dimensional sections F ∈ G n , k with outer volume ratio bounded by ovr ( K ∩ F ) ⩽ C n n − k log ( 1 + n n − k ) .