2019/10/14 by Emanuel Milman, Milman, Emanuel, Yuval Yifrach +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Point processes and geometric inequalities #math.FA
paper · pdf · doi:10.48550/arxiv.1910.06033
17 pages; final version, to appear in J. Func. Anal
openalex publication_date 2019/10/14 · arxiv created 2021/05/26 · arxiv updated 2021/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.