.  We use the min-max inequality to get $$\begin{aligned} \varphi^0(\beta)  &= \max_{z\in {\mathcal{B}}^\star}\, \min_{g\in \partial\|\beta\|} ~{\|g-z\|}_0^0 \nonumber \\ &\leq \min_{g\in \partial\|\beta\|} \,\max_{z\in {\mathcal{B}}^\star} ~{\|g-z\|}_0^0 \nonumber \\ &= \min_{g\in \partial\|\beta\|} \,\max_{z\in {\mathcal{B}}^\star} ~{\|g+z\|}_0^0 \nonumber  $$ where we used the symmetry of ${\mathcal{B}}^\star$. We now focus on the inner optimization problem. 