Solution
= Solution
After the replacement, the objective is
$$
\sup_{\|v\|_2\leq1}|(\mu_0-\mu)^Tv|
=\|\mu-\mu_0\|_2
$$
by <Euclidean norm duality>. It is nonnegative and equals zero exactly at $\mu=\mu_0$, so $\mu_0$ is the unique minimizer.
= Solution
After the replacement, the objective is
$$
\sup_{\|v\|_2\leq1}|(\mu_0-\mu)^Tv|
=\|\mu-\mu_0\|_2
$$
by <Euclidean norm duality>. It is nonnegative and equals zero exactly at $\mu=\mu_0$, so $\mu_0$ is the unique minimizer.