Applications of Convex Analysis to the Smallest Intersecting Ball Problem

Abstract

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

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• A Generalized Sylvester Problem and a Generalized Fermat-Torricelli Problem
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• Minimal time functions and the smallest intersecting ball problem generated by unbounded dynamics
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• Subgradient Algorithm, Stochastic Subgradient Algorithm, Incremental Subgradient Algorithm, and Set Location Problems