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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 $p \to \infty$?

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You can reduce things a bit as follows. Suppose your two vectors are (a,b,c) and (x,y,z). Without loss of generality all of a, b and c are non-negative. Let's suppose that a is bigger than both b and c. Pick a third vector (u,v,w) orthogonal to both (a,b,c) and (x,y,z). If u is non-zero, then we can add a tiny multiple of (u,v,w) to (a,b,c) in such a way that the $\ell_2$ norm increases by a quadratic amount but the $\ell_\infty$ norm decreases by a linear amount. So after rescaling we have got a better example.

I won't go through all possible cases here, but we now know that either u is zero or (WLOG) a=b. The same proof also tells us that the maximum absolute value of x, y and z is attained twice. Probably it's fairly easy to show, after a little bit of case analysis, that this maximum is attained once for a positive value and once for a negative value, and in places 2 and 3. If that works, then we've got to (a,a,b) and (x,-y,y), with a bigger than b and x bigger than y in absolute value. This starts to look like a much more manageable optimization problem.

I don't guarantee that all this works, but I think it has a good chance.

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