2024/10/19 by Stanisław J. Szarek, Szarek, Stanislaw, Paweł Wolff +1
Engineering · Mathematics · #46B20 #52A40 #68P30 #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2410.15118
openalex publication_date 2024/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The celebrated Dvoretzky theorem asserts that every N-dimensional convex body admits central sections of dimension d = Ω(log N), which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to some norm, much better lower bounds on d have been obtained, with most research focusing on such lower bounds and on the degree of approximation of the section by a d-dimensional Euclidean ball. In this note we concentrate on another parameter, namely the radius of the approximating ball. We focus on the case of the unit ball of the space ℓ1N (the so-called cross-polytope), which is relevant to various questions of interest in theoretical computer science. We will also survey other instances where similar questions for other normed spaces (most often ℓp-spaces or their non-commutative analogues) were found relevant to problems in various areas of mathematics and its applications, and state some open problems. Finally, in view of the computer science ramifications, we will comment on the algorithmic aspects of finding nearly spherical sections.