On random intersections of two convex bodies. Appendix to: "Isoperimetry   of waists and local versus global asymptotic convex geometries" by   R.Vershynin
  In the paper "Isoperimetry of waists and local versus global asymptotic convex geometries", it was proved that the existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of randomly rotated K and L is nicely bounded. In this appendix, we achieve a polynomial bound on the diameter of that intersection (in the ratio of the dimensions of the sections). 
Mark Rudelson and Roman Vershynin