Minimizing $\max | x_i - \mu |$
Since $|x| = \max (x, -x)$ for any $x \in \mathbb{R}$, we have $|x_i - \mu| = \max(x_i - \mu, \, \mu - x_i)$. Therefore,
$$
\max_i |x_i - \mu| = \max \left(\max_i (x_i - \mu), \, \max_i (\mu - x_i) \right) = \max (g - \mu, \, \mu - l),
$$
where $g = \max_i x_i$ and $l = \min_i x_i$.
Finally, the function $\max(g - \mu, \mu - l)$ has minimum at $\mu = \frac{l + g}{2}$.