The "largest square in a cube" problem, which asks for the largest square inside a cube, has a solution as can be seen on this page, which also says that the general problem in higher dimensions is unsolved.

The problem is equivalent to the following optimization problem: find two orthogonal unit vectors in $\mathbb{R}^3$ such that the maximum of the absolute values of all their coordinates is minimized. For the "largest square in a cube" problem to have the answer it does, this minimum should be $2/3$, i.e., it occurs when the coordinates are $(2/3,2/3,1/3),(1/3,-2/3,2/3)$ (or some coordinate permutation-cum-axis reflection of those). How would we show that this is indeed where the optimum occurs? This seems like it should be some really simple algebra, but it is eluding me.

More abstractly, this is asking for a pair of orthogonal unit vectors such that the maximum of their $\ell^\infty$-norms is minimized. What happens if we are trying to minimize the larger of the $\ell^p$-norms, $p > 2$? Does the minimizing value tend to $2/3$ as $p \to \infty$?