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

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.

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“…To guarantee that we can sample treatment vector subject to the optimized covariance, we first present a lemma that is equivalent to the Grothendieck's identity [28].…”

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