I'm interested in the question of finding the maximum area of $A\subset S^{d-1}$, such that, for all $x,y \in A, \left<x,y\right>\ge 0$. The portion of the sphere lying in the positive orthant seems like a reasonable guess. Can any subset of the sphere that satisfies the positive inner product property be rotated to fit in the positive orthant? I seem to have some evidence that the answer to this last question is "no" but I would like to know for certain.