Let $(X,\mathscr{F},\mu)$ be a $\sigma$-finite measure space, $u$ be a strictly concave function from $\mathbb{R}$ to $\mathbb{R}$. How can one (constructively) find the maximum of $$ \operatorname{sup}_{f\in B}\int_{x \in X} u(f(x)) dx , $$ where $B$ is the unit ball in $L^p(\mu)$ without the ($\mu$-a.e.) constant functions, equipped with the weak-topology.
It does not seem to be obvious, even the simplification I made to $u(x)=log(x^2)$ gives me grief.