functional $G_h$ is twice Fréchet differentiable, with gradient $$\begin{aligned}
\nabla G_h (V) &= - {{\mathbb{E}}\left[ \frac{(X- h)(X- h)^T - V}{{\left\| (X- h)(X- h)^T - V \right\|}_F}\right]}.
\label{def:gradV}\end{aligned}$$ and Hessian operator, $ \nabla _h^2 G(V) : \mathcal{S}(H) \to \mathcal{S}(H)$, $$\begin{aligned}
\nabla _h^2G (V) &=  {{\mathbb{E}}\left[ \frac{1}{{\left\| Y(h) - V \right\|}_F}\left( I_{S(H)} - \frac{(Y(h) - V) \otimes_F  (Y(h) - V) }{{{\left\| Y(h) - V \right\|}_F}^2} \right)\right]}.
\label{def:HeV}\end{aligned}$$ where $Y(h) = (X-h)(X-h)^T$, $I_{S(H)}$ is the identity operator on $\mathcal{S}(H)$ and $A \otimes_F B (V) = \langle A, V \rangle_F B$ for any elements $A, B$ and $V$ belonging to $\mathcal{S}(H)$.

Furthermore, $\Gamma_m$ is also defined as the unique zero of the non linear equation: $$\begin{aligned}
\nabla G_m (\Gamma_m) &= 0.
\label{def:zeroV}\end{aligned}$$ Remarking that previous equality can be rewritten as follows, $$\begin{aligned}
\Gamma_m &=  \frac{1}{{{\mathbb{E}}\left[\frac{1}{ {\left\|  (X- m)(X-m)^T - \Gamma_m \right\|}_F} \right]}}{{\mathbb{E}}\left[\frac{ (X-m)(X-m)^T }{ {\left\|  (X- m)(X-m)^T - \Gamma_m \right\|}_F}\right]},
\label{def:baseweiszfled}\end{aligned}$$ it is clear that $\Gamma_m$ is a bounded, symmetric and non negative operator in $\mathcal{S}(H)$.

As stated in Proposition 2 of [@KrausPanaretos2012], operator $\Gamma_m$ has an important stability property when the distribution of $X$ is symmetric, with finite second moment, *i.e* ${{\mathbb{E}}\left[{\left\| X \right\|}^2\right]}< \infty$. Indeed, the covariance operator of $X$, $\Sigma = 