  Existence of nicely bounded sections of two symmetric convex bodies K and L
implies that the intersection of random rotations of K and L is nicely bounded.
For L = subspace, this main result immediately yields the unexpected
phenomenon: "If K has one nicely bounded section, then most sections of K are
nicely bounded". This 'existence implies randomness' consequence was proved
independently in [Giannopoulos, Milman and Tsolomitis]. The main result
represents a new connection between the local asymptotic convex geometry (study
of sections of convex bodies) and the global asymptotic convex geometry (study
of convex bodies as a whole). The method relies on the new 'isoperimetry of
waists' on the sphere due to Gromov.
