Regular random sections of convex bodies and the random Quotient-of-Subspace Theorem

It was shown by G. Pisier that any finite-dimensional normed space admits an α-regular M-position, guaranteeing not only regular entropy estimates but moreover regular estimates on the diameters of minimal sections of its unit-ball and its dual. We revisit Pisier's argument and show the existence of a different position, which guarantees the same estimates for randomly sampled sections with high-probability. As an application, we obtain a random version of V. Milman's Quotient-of-Subspace Theorem, asserting that in the above position, typical quotients of subspaces are isomorphic to Euclidean, with a distance estimate which matches the best-known deterministic one (and beating all prior estimates which hold with high-probability). Our main novel ingredient is a new position of convex bodies, whose existence we establish by using topological arguments and a fixed-point theorem.