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Point on sphere greatest distance from given points

How can I use Mathematica to solve this problem:

Given n points on a sphere, I want to find a point x such that:


is maximal, where "distance" is spherical ("great circle") distance.

In other words, I want to find the point x "furthest" from the given n points.

Is there any chance x will coincide with one of the given points? If
so, is there a better notion of distance to use?

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