I have seen the expectation of a random vector expressed as the solution to the optimization problem:
\begin{equation}
\mathbb{E}[X]=argmax_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= \int_{\Omega} \|X(\omega)-v\|^2 dP(\omega)).
\end{equation}


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My question is... can we express the covariance matrix of a random vector as a similar optimization problem?