\begin{equation}
	\begin{aligned}
		&|d_{\mathcal{M}_d}(\boldsymbol{H+\epsilon}) - d_{\mathcal{M}_d}(\boldsymbol{H})| 
		\leq\Vert \boldsymbol{\epsilon} -\boldsymbol{1}{(\boldsymbol{x}^{\boldsymbol{H+\epsilon}}_{\min}-\boldsymbol{x}^{\boldsymbol{H}}_{\min})}^\top \Vert_F \\
		\leq &d_{\mathcal{M}_d}(\boldsymbol{\epsilon})= \Vert(\boldsymbol{I}-\boldsymbol{ee}^\top )\boldsymbol{\epsilon}\Vert_F 
		= \Vert \boldsymbol{\epsilon} -\boldsymbol{1}{\boldsymbol{x}^{\boldsymbol{\epsilon}}_{\min}}^\top\Vert_F 
		\leq  \Vert \boldsymbol{\epsilon} \Vert_F.
	\end{aligned}
\end{equation}
