often called the geometric median of $X$. It is uniquely defined under broad assumptions on the distribution of $X$ (see [@Kem87]) which can be expressed as follows.

\[eq:supportCdtnmed\] There exist two linearly independent unit vectors $(u_1,u_2) \in H^2$, such that $${\mathbf{Var}}( {\left\langle u,X \right\rangle} ) > 0, \quad \mbox{for }u \in \{u_1,u_2\} .$$

If the distribution of $X-m$ is symmetric around zero and if $X$ admits a first moment that is finite then the geometric median is equal to the expectation of $X$, $m = {{\mathbb{E}}\left[X\right]}$. Note however that the general definition (\[defmed\]) does not require to assume that the first order moment of ${\left\| X \right\|}$ is finite since $| {{\mathbb{E}}\left[ {\left\| X-u \right\|} - {\left\| X \right\|}\right]} | \leq {\left\| u \right\|}$.

The (geometric) median covariation matrix (MCM)
-----------------------------------------------

We now consider the special vector space, denoted by $\mathcal{S}(H)$, of $d \times d$ matrices when $H= \mathbb{R}^d$, or for general separable Hilbert spaces $H$, the vector space of linear operators mapping $H \to H$. Denoting by $\{e_j, j \in J \}$ an orthonormal basis in $H$, the vector space $\mathcal{S}(H)$ equipped with the following inner product: $$\begin{aligned}
\langle A, B \rangle_F &= \sum_{j \in J} \langle A e_j, 