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!

  • $\begingroup$ If you have two antipodal points, construct a great circle containing both. The closed hemispheres defined by these great circles both contain the two antipodal points. $\endgroup$ – ncmathsadist Dec 5 '18 at 15:16

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