Definition: A matrix $ C in mathbb R ^ {m times n} $ is a local correlation matrix if there are true random variables $ x_1, dots, x_m, y_1, dots, y_n $ defined on a common probability space in which values are recorded $[1,+1]$ so that $$ C_ {ij} = mathbb E[x_iy_j]$$ holds with everyone $ (i, j) in {1, dots, m } times {1, dots, n } $,
Set of all $ m times n $ local correlation matrices $ mathsf {LC} _ {m, n} $,

What is the measure of $ mathsf {LC} _ {m, n} cap[1,+1]^ {m times n} $ in the $[1,+1]^ {m times n} $?

Is there a nontrivial geometric or functional characterization of $ mathsf {LC} _ {m, n} $?

Are there explicit template families in $[1,+1]^ {m times n} $ they are not$ mathsf {LC} _ {m, n} $?

Given a $ m times n $ Matrix, there is a simple test if it is a member $ mathsf {LC} _ {m, n} $?