Whilst doing an exercise, on bounds on total absolute curvature I encountered the following obstacle:
Let $ \mathit c$ be a closed curve, whose trace is contained in $\mathbb S^2$, if c has no antipodal points, does this imply, that c is contained in some open hemisphere? Intuitively I would argue, that if $ \pmb P, Q$ are the points on the curve that are the furthest apart, one takes the great circle parallel to the line connecting the two points, and then the curve should be contained, in one of the open hemispheres whose equator is formed by this great circle. I'm not sure if the claim is even true, and if yes how to make it rigorous. Any help would be greatly appreciated.