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I need to refer to the "top" of a hemisphere - the "highest point" on a hemisphere. I am thinking it must be called the "apex" of the hemisphere, but I am not sure.

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North pole?${}$ – David Mitra Jul 29 '12 at 2:08
Agreed, David. It is picturesque and entirely accurate. – ncmathsadist Jul 29 '12 at 2:11
I asked an Australian once if they orient their globes with the South Pole uppermost, but he said no, they orient them the same way that everyone else does. – MJD Jul 29 '12 at 3:11
Just plain "pole" ought to be fine... – J. M. Jul 29 '12 at 3:31
il colmo dellagran secca… – Michael Jul 29 '12 at 3:47

The term Chebyshev center is well established, despite the confusing "alternative" definition in the first paragraph of the article. A Chebyshev center of a set $A$ in metric space $X$ is a point $c\in X$ (which may or may not be unique) which minimizes $\sup_{a\in A} d(a,c)$. If $X$ is a hemisphere with either extrinsic (chordal) or intrinsic (Riemannian) metric, the Chebyshev center of $X$ is the point you want to describe. This description is less intuitive than North Pole, but is invariant under rigid motions.

(Aside: a very nice application of Chebyshev centers to a fixed point problem.)

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Perhaps I can simply refer to it as the "center" of the hemisphere. – bobobobo Jul 30 '12 at 11:57
@bobobobo But that may be confused with the geometric center of the sphere. On second thought: Chevyshev center is what I'd use for a generic set, but for this specific shape pole is better. You can define it in a few words even without imposing a particular orientation of the hemisphere: the pole is the intersection of hemisphere with its axis of symmetry. – user31373 Jul 30 '12 at 12:25

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