The Optimal Multilinear Bohnenblust–Hille Constants: A Computational Solution for the Real Case. (18th November 2018)

• Record Type: Journal Article
• Authors: Vieira Costa Júnior, Fernando
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 15(2018)
• Page Start: 1656
• Page End: 1668
• Publication Date: 2018-11-18
• Subjects: Bohnenblust–Hille inequality -- optimal constants; 15-04 -- 46-04; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1489414
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: The Bohnenblust–Hille inequality for m-linear forms was proven in 1931 as a generalization of the famous 4/3-Littlewood inequality. The optimal constants (or at least their asymptotic behavior as m grows) is unknown, but significant for applications. A recent result, obtained by Cavalcante, Teixeira and Pellegrino, provides a kind of algorithm, composed by finitely many elementary steps, giving as the final outcome the optimal truncated Bohnenblust–Hille constants of any order. But the procedure of Cavalcante et al. has a fairly large number of calculations and computer assistance cannot be avoided. In this paper we present a computational solution to the problem, using the Wolfram Language. We also use this approach to investigate a conjecture raised by Pellegrino and Teixeira, asserting thatC m = 2 1 − 1 / m for all m ∈ ℕ and to reveal interesting unknown facts about the geometry ofB L ( 3 ℝ 3 ) .