We set $D = (d_{ij})_{(i,j) \in \llbracket 1,n\rrbracket^2} = (\sqrt{m_{ij}})_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{M}_n(\mathbb{R})$ and $M_c = \left((d_{ij} + c\xi_i^j)^2\right)$ with $c > 0$. We seek the minimal constant $c^* > 0$ (if it exists) satisfying:
\begin{itemize}
  \item $\Psi(M_{c^*})$ has non-negative eigenvalues,
  \item for all $c > c^*$ and for all non-zero vector $X \in \mathcal{H}$, ${}^t X \Psi(M_c) X > 0$.
\end{itemize}
We know that $c^*$ is bounded above by $\widetilde{c}$.

We consider $\mathcal{A} = \left\{X \in \mathcal{H} \mid \|X\| = 1 \text{ and } 4\left({}^t X \Psi(D) X\right)^2 - 2\, {}^t X \Psi(M) X \geqslant 0\right\}$ and we define the mapping
$$\alpha: \begin{cases}\mathcal{A} \longrightarrow \mathbb{R} \\ X \longmapsto -2\, {}^t X \Psi(D) X + \sqrt{4\left({}^t X \Psi(D) X\right)^2 - 2\, {}^t X \Psi(M) X}\end{cases}$$

Show that there exists $X^* \in \mathcal{A}$ such that $\alpha(X^*) = \sup_{X \in \mathcal{A}} \alpha(X)$ and $\alpha(X^*) > 0$.