Assume $A\subset S^2$ a closed subset. I am interested in necessary and sufficient conditions for $A$ to lie in a (open or closed hemisphere).
For example, it is necessary for $A$ to not contain antipodal points. And given this condition I think it is possible to show that if it contained in a closed hemisphere then it is contained in an open one.
Furthure more, a quite strong condition - the diameter being less than $\pi/2$ will suffice, although It is not necessary (think of a thin strip contained in the upper hemisphere and almost escaping it from both sides.
Another for a sufficient condition is (spherical) convexity, though of course it is not necessary.
I suspect that Borsuk-Ulam might be of help.
I ask if anyone has an idea or knows a reference? thank you!