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Does there exist a ''universal'' constant, $c > 0$ say, such that for any(!) $k \in \mathbb{N}$ every(!) $k \times k$-correlation matrix $\Sigma$ can be written as $\Sigma = c\Theta$, where $\Theta$ is an element of the absolute convex hull of $k \times k$-correlation matrices of $\underline{rank \hspace{0.1cm} 1}$?

I guess there exists a counterexample. However, I haven't found one so far. Any hints and links to suitable references are highly welcome!

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