- The slicing problem can be reduced to the study of isotropic convex bodies K with diam(K) cnLK, where LK is the isotropic constant. We study the 2-behaviour of linear functionals on this class of bodies. It is proved that ||,||2 CLK for all in a subset U of Sn-1 with measure (U) 1 - exp(-cn). However, there exist isotropic convex bodies K with uniformly bounded geometric distance from the Euclidean ball, such that maX Sn-1 ||,||2 c4nLK. In a different direction, we show that good average 2-behaviour of linear functionals on an isotropic convex body implies very strong dimension-dependent concentration of volume inside a ball of radius r nLK.