2024/09/10 by Hermann König, König, Hermann
Mathematics · #46B07 #52A38 #52A40 #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2409.06432
openalex publication_date 2024/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
The maximal hyperplane section of the l_∞n-ball, i.e. of the n-cube, is the one perpendicular to 1/sqrt 2 (1,1,0, ... ,0), as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the lpn-balls for very large p ≥ 1015. By Oleszkiewicz, Ball's result does not transfer to lpn for 2 < p < p0 ≃ 26.265. Then the hyperplane section perpendicular to the main diagonal yields a counterexample for large dimensions n. We show that the analogue of Ball's result holds in lpn-balls for all hyperplanes with normal unit vectors a, if all coordinates of a have modulus ≤ \frac 1 √ 2 and p has distance ≥ 2-p to the even integers. Under similar assumptions, we give a Gaussian upper bound for 20 < p < p0.