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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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    @Mark, I guess I really meant the implicit assumption that the hemisphere is the northern one. The question does say "top", but ideally the answer should be independent of whether the hemisphere is placed with the convex part facing up, down, or to the side...2012-07-30

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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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    @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.2012-07-30