Grothendieck's inequality and completely correlation preserving functions -- a summary of recent results and an indication of related research problems

TL;DR

This paper reviews recent progress on Grothendieck's inequality, focusing on functions preserving correlation matrices and outlining open problems and future research directions in this challenging area.

Contribution

It introduces new approaches using Schur products, Gaussian analysis, and copulas to study the Grothendieck constant and summarizes recent findings and open problems.

Findings

New bounds and approaches for the Grothendieck constant

Identification of functions preserving correlation matrices

Open problems for future research in correlation preservation

Abstract

As part of the search for the value of the smallest upper bound of the best constant for the famous Grothendieck inequality, the so-called Grothendieck constant (a hard open problem - unsolved since 1953), we provide a further approach, primarily built on functions which map correlation matrices entrywise to correlation matrices by means of the Schur product, multivariate Gaussian analysis, copulas and inversion of suitable Taylor series. We summarise first results and point towards related open problems and topics for future research.