\in \mathcal{S}(H)$. The second term at the right-hand side of (\[def:popriskcov\]) prevents from having to introduce hypotheses on the existence of the moments of $X$. Introducing the random variable $Y := (X- m)(X- m)^T$ that takes values in $\mathcal{S}(H)$, the MCM is unique provided that the support of $Y$ is not concentrated on a line and Assumption 1 can be rephrased as follows in $\mathcal{S}(H)$,

\[eq:supportCdtnCov\] There exist two linearly independent unit vectors $(V_1,V_2) \in \mathcal{S}(H)^2$, such that $${\mathbf{Var}}( {\left\langle V,Y \right\rangle}_F ) > 0, \quad \mbox{for }V \in \{V_1,V_2\} .$$

We can remark that Assumption \[eq:supportCdtnmed\] and Assumption \[eq:supportCdtnCov\] are strongly connected. Indeed, if Assumption \[eq:supportCdtnmed\] holds, then ${\mathbf{Var}}( {\left\langle u,X \right\rangle} ) > 0$ for $ u \in \{u_1, u_2\}$. Consider the rank one matrices $V_1 = u_1u_1^T$ and $V_2 = u_2 u_2^T$, we have ${\left\langle V_1,Y \right\rangle}_F = \langle u_1, X-m \rangle^2$ which has a strictly positive variance when the distribution of $X$ has no atom. More generally $ {\mathbf{Var}}({\left\langle V_1,Y \right\rangle}_F) >0$ unless there is a scalar $a >0$ such that $\mathbb{P}\left[ \langle u_1, X-m \rangle = a\right] = \mathbb{P}\left[ \langle u_1, X-m \rangle = -a\right] = \frac{1}{2}$ (assuming also that $\mathbb{P}\left[  X-m = 0\right] = 