[Paper Review] Geometry of the closed unit ball of the space of bilinear forms on $\ell_{\infty}^{2}$

W. Cavalcante, Daniel Pellegrino|arXiv (Cornell University)|Mar 4, 2016

Advanced Harmonic Analysis Research3 citations

TL;DR

This paper fully characterizes the extreme and exposed points of the closed unit ball in the space of real-valued bilinear forms on $\ell_{\infty}^2$. It proves that all bilinear forms achieving the optimal constant in Littlewood's $4/3$ inequality are extreme points, and provides analytical and numerical evidence suggesting the optimal constant for complex bilinear forms with real coefficients is trivially 1.

ABSTRACT

We obtain all extreme and exposed points of the closed unit ball of the space of bilinear forms $T:\ell_{\infty}^{2} imes\ell_{\infty}^{2} ightarrow \mathbb{R}.$ We also show that any (norm one) bilinear form $T:\ell_{\infty }^{2} imes\ell_{\infty}^{2} ightarrow\mathbb{R}$ for which the optimal constant of the Littlewood's $4/3$ inequality is achieved is necessarily an extreme point. In the case of complex scalars we combine analytical and numerical evidences supporting that, at least for complex bilinear forms with real coefficients, the optimal constant of Littlewood's $4/3$ inequality seems to the the trivial, i.e., $1.$

Motivation & Objective

• To completely classify all extreme and exposed points of the closed unit ball in the space of real bilinear forms on $\ell_{\infty}^2$.
• To identify all bilinear forms in the unit ball that achieve the optimal constant in Littlewood's $4/3$ inequality.
• To investigate whether the optimal constant for complex bilinear forms with real coefficients is trivially 1, using analytical and numerical evidence.

Proposed method

• Derives explicit norm expressions for real and complex bilinear forms on $\ell_{\infty}^2$ using trigonometric parameterizations of unit vectors.
• Applies optimization techniques to maximize the sum of absolute values of bilinear form coefficients under $\ell_\infty$-norm constraints.
• Uses symmetry and case analysis to classify extreme and exposed points based on coefficient configurations.
• Employs analytical and numerical methods to evaluate the $L^{4/3}$-norm of bilinear forms and compare them to the operator norm.
• Applies lemmata on inequalities involving absolute values and signs of coefficients to establish necessary and sufficient conditions for norm equality.
• Analyzes the ratio between the $L^{4/3}$-norm and the operator norm to determine when equality (i.e., optimality) is achieved.

Experimental results

Research questions

• RQ1 Which bilinear forms on $\ell_{\infty}^2$ are extreme or exposed points of the closed unit ball in the space of real bilinear forms?
• RQ2 What is the precise set of real bilinear forms in the unit ball that achieve the optimal constant in Littlewood's $4/3$ inequality?
• RQ3 Is the optimal constant in Littlewood's $4/3$ inequality for complex bilinear forms with real coefficients equal to 1, as suggested by analytical and numerical evidence?

Key findings

• All extreme and exposed points of the closed unit ball in $\mathcal{L}(^2\ell_{\infty}^2(\mathbb{R}))$ are completely characterized and correspond to specific coefficient configurations of the bilinear form.
• Any bilinear form achieving the optimal constant in Littlewood's $4/3$ inequality on $\ell_{\infty}^2$ with real coefficients is necessarily an extreme point of the unit ball.
• The optimal constant in Littlewood's $4/3$ inequality for real bilinear forms on $\ell_{\infty}^2$ is $\sqrt{2}$, and this value is achieved precisely by forms with coefficients satisfying $|a_{11}+a_{21}|+|a_{12}+a_{22}|=\sqrt{2}$ and $|a_{11}-a_{21}|+|a_{12}-a_{22}|=\sqrt{2}$.
• For complex bilinear forms with real coefficients, analytical and numerical evidence strongly supports that the optimal constant in Littlewood's $4/3$ inequality is trivially 1.
• The paper identifies four canonical forms (up to sign and permutation) that achieve the optimal constant in the real case, such as $T(x,y) = \alpha z_1w_1 + \alpha z_1w_2 + \alpha z_2w_1 - \alpha z_2w_2$.
• The condition $\left|\frac{cd}{ab}(a-b)\right| \leq c-d$ and $\left|\frac{ab}{cd}(c+d)\right| \leq a+b$ is both necessary and sufficient for certain norm expressions to coincide, which underpins the classification of extreme points.