Within some points on a 2D-plane, find one point that is less than "P" away from each point.

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If there are 10 given points on a 2d plane, how would one find a final point who's distance is less than P in distance away from each given point? Assuming such a final point exists, a final point that is less than P away from any given point. Is there an existing linear programming problem similar to this one might study?