Journal of Convex Analysis 19 (2012), No. 2, 497--518
Copyright Heldermann Verlag 2012
Applications of Convex Analysis to the Smallest Intersecting Ball Problem
Nguyen Mau Nam
Dept. of Mathematics, The University of Texas - Pan American, Edinburg, TX 78539--2999, U.S.A.
nguyenmn@utpa.edu
Thai An Nguyen
Dept. of Mathematics, Hue University, 32 Leloi Hue, Vietnam
thaian2784@gmail.com
Juan Salinas
Dept. of Mathematics, The University of Texas - Pan American, Edinburg, TX 78539--2999, U.S.A.
jsalinasn@broncs.utpa.edu
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 1857 by J. J. Sylvester. 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.
Keywords: Convex analysis, convex optimization, generalized differentiation, smallest enclosing ball problem, smallest intersecting ball problem, subgradient-type algorithms.
MSC: 49J52, 49J53, 90C31
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