2024/03/21 by Hermann König, König, Hermann
Engineering · Mathematics · #46B07 #52A38 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2403.14456
openalex publication_date 2024/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For 2 < p < p0 ≃ 26.265, the hyperplane section of the lpn-unit ball Bpn perpendicular to a^(n) = 1/sqrt(n) (1, ... ,1) for large n has larger volume than the one orthogonal to a^(2) = 1/sqrt(2) (1,1,0, ...,0), as shown by Oleszkiewicz. This is different from the case of l_∞n considered by Ball. We give a quantitative estimate for which dimensions n this happens, namely for n > c (\frac 1 p0-p + \frac 1 p-2) for some absolute constant c>0. Correspondingly for projections of Bqn onto hyperplanes, Barthe and Naor showed that projections onto hyperplanes perpendicular to a(n) have smaller volume for large n than onto the one orthogonal to a(2), if \frac 4 3 < q < 2, different from the case q=1. We show that this happens for all n > 5 (\frac 1 q-\frac 4 3 + \frac 1 2-q).