The context of this question is the step to go from (24) to (25) here.
One considers an optimization problem of the following form $$\inf_{\rho \in S}\sup_{U}\ (\cdot)$$
where $S$ is a convex set and $U$ is a unitary matrix such that $\|U\|_\infty\leq 1$. It is claimed that the supremum is taken over a convex set and hence, one can use a minimax theorem to commute the $\inf$ and the $\sup$ i.e.
$$\inf_{\rho \in S}\sup_{U}\ (\cdot) = \sup_{U}\inf_{\rho \in S}\ (\cdot)$$
I don't understand why the set
$$\{U\ \vert \ U^\dagger U = I, \|U\|_\infty\leq 1\}$$
is convex. A linear combination of unitaries is not a unitary. So why do they claim that the variation is over a convex set and why can one use a minimax theorem here?
They claim something different: First, they solve the inner problem and compute the sup. Then they observe that the inner problem can be replaced by taking the supremum over $\{U: \ \|U\|_\infty\le 1\}$, which is convex. So no unitary matrices involved anymore.