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.

G. Pisier 表明，任何有限维赋范空间都存在α -正则M -位置，不仅保证了正则熵估计，而且还保证了对其单位球及其对偶的最小截面直径的正则估计。我们重新审视 Pisier 的论点并展示了不同位置的存在，这保证了对具有高概率的随机采样部分的相同估计。作为应用，我们获得了V. Milman 子空间商数定理的随机版本，断言在上述位置，典型的子空间的商与欧几里得是同构的，其距离估计与最著名的确定性相匹配（并击败所有以高概率成立的先前估计）。我们的主要新颖成分是凸体的新位置，我们通过使用拓扑参数和不动点定理来确定其存在性。