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

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 4 / a / Solution

{title2=$\sup_{\lVert v\rVert_2\leq1}|x^Tv|=\lVert x\rVert_2$}

The Euclidean norm is self-dual:
$$
\sup_{\lVert v\rVert_2\leq1}|x^Tv|=\lVert x\rVert_2.
$$
The upper bound is <Cauchy-Schwarz inequality>, and equality holds at $v=x/\lVert x\rVert_2$ when $x\ne0$.

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 Incoming links: Euclidean norm duality