The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound

Abstract

The Grothendieck constant $K_{G}$ is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of $K_{G}$ is unknown. The best known lower bound on $K_{G}$ was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that $K_{G} \ge K_{DR} + 10^{-12}$, where $K_{DR}$ denotes the Davie-Reeds lower bound. Our argument is based on a perturbative analysis of the Davie-Reeds operator. We show that every near-extremizer for the Davie-Reeds problem has $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.

Figures (2)

• Figure 1: One-dimensional "square wave" examples of Davie--Reeds optimizers. In each example, $f=g=\mathop{\mathrm{sign}}\nolimits(x)$ outside the strip $[-C^*,C^*]$, while $g=-f$ inside the strip, and the breakpoints are chosen so that $\mathop{\mathrm{\mathbb{E}}}\limits_{x \sim {\cal N}(0,1)} xf(x) \bm{1}_{|x| \leq C^*} = 0$.
• Figure 2: Plots of $F(C) := 4\phi(C)^2 - \lambda^* (4\Phi(-C) - 1)$ along with its first derivative, for $\lambda^* \approx 0.19748$. The only two roots of $F'$ are $C_- \approx 0.25573$ and $C_+ \approx 2.0582$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 2.1: pisier2012grothendieck
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1
• proof