2020/09/27 by Tomasz Kobos, Kobos, Tomasz · 1 citation
Mathematics · #41A65 #47A30 #52A21 #52A22 #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Point processes and geometric inequalities #Primary 47A58
paper · pdf · doi:10.48550/arxiv.2009.12929
openalex publication_date 2020/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a centrally symmetric spherical and simplicial polytope, whose vertices form a (1)/(4n)-net in the unit sphere in ℝn. We prove a uniform lower bound on the norms of all hyperplane projections P: X → X, where X is the n-dimensional normed space with the unit ball K. The estimate is given in terms of the determinant function of vertices and faces of K. In particular, if N ≥ n4n and K = \conv \ ± x1, ± x2, …, ± xN \, where x1, x2, …, xN are independent random points distributed uniformly in the unit sphere, then every hyperplane projection P: X → X satisfies an inequality ‖P‖X ≥ 1+cnN-(2n2+4n+6) (for some explicit constant cn), with the probability at least 1 - (3)/(N).