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The smallest enclosing circle problem asks for the circle of smallest radiusenclosing a given set of finite points on the plane. This problem wasintroduced in the 19th century by Sylvester [17]. After more than a century,the problem remains very active. This paper is the continuation of our effortin shedding new light to classical geometry problems using advanced tools ofconvex analysis and optimization. We propose and study the followinggeneralized version of the smallest enclosing circle problem: given a finitenumber of nonempty closed convex sets in a reflexive Banach space, find a ballwith the smallest radius that intersects all of the sets.

Applications of Convex Analysis to the Smallest Intersecting Ball Problem