1. On the $L^\infty$ calculus of variations:
The field known as the $L^\infty$ calculus of variations is a relatively new field that concerns itself with minimising functionals involving the supremum norm.
One often obtains $L^\infty$ variational problems as “limits” of $L^p$ variational problems as $p \to \infty$. As a canonical example, the problem of minimising the functional $\int |\nabla u|^p$ leads to the variational problem of minimising the supremum norm of $\nabla u$ in the limit as $p \to \infty$. We refer to the set of notes by Lindqvist for a complete exposition of the limiting process, and more.
Minimisers of the supremum norm of $\nabla u$ satisfy the so called infinity Laplace equation, given by
$$\Delta_\infty u := \sum_{i, j} \frac{\partial^2 u}{\partial x_i \partial x_j} \frac{\partial u}{\partial x_i} \frac{\partial u}{\partial x_j} = 0.$$
2. A strange minimisation problem:
Consider now the variational problem of minimising the functional
$$F(h) := \limsup_{i \to \infty} \|f_i - h\|_{L^1}$$
for some fixed sequence of functions $f_i \in L^1([0, 1])$, over all $h \in L^1([0, 1])$.
This somewhat strange looking variational problem happens to have a rather delicate proof, courtesy of user fedja. As observed in the comments by Pietro Majer, the proof seems to encode a proof of Komlos’ theorem, a powerful result that implies as an immediate corollary and special case, the $L^1$ strong law of large numbers.
I had at some point also convinced myself that a proof of Komlos’ theorem was hidden in the given proof, but I have long since forgotten my reasoning.
3. Taking the $L^\infty$ limit:
Motivated by the usual scheme of the $L^\infty$ calculus of variations, we wish to take the limit, as $p \to \infty$ of the problem of minimising
$$F_p (h) := \limsup_{i \to \infty} \|f_i - h\|_{L^p}$$
over all $h \in L^p ([0, 1])$. By a heuristic argument that is too complicated to write here, I guessed the following form of the $p \to \infty$ limit. Unfortunately, as discussed in the comments by Pietro Majer and myself, this formulation seems to miss the mark and leads to a not very interesting answer.
Question: What is the correct limit, as $p \to \infty$ of the problem of minimising
$$\limsup_{i \to \infty} \|f_i - h\|_{L^p},$$
and what information does the variational problem encode?
