Journal of Convex Analysis 18 (2011), No. 2, 589--600
Copyright Heldermann Verlag 2011
Validity of the Union of Uniform Closed Balls Conjecture
Chadi Nour
Dept. of Computer Science and Mathematics, Lebanese American University, P. O. Box 36, Byblos, Lebanon
cnour@lau.edu.lb
Ron J. Stern
Dept. of Mathematics and Statistics, Concordia University, 1400 De Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada
stern@mathstat.concordia.ca
Jean Takche
Dept. of Computer Science and Mathematics, Lebanese American University, P. O. Box 36, Byblos, Lebanon
jtakchi@lau.edu.lb
A proof is provided for the union of uniform closed balls conjecture introduced in a previous paper of the authors ["Proximal smoothness and the exterior sphere condition", J. Convex Analysis 16 (2009) 501--514]; see also C. Nour, R. J. Stern and J. Takche, "The union of uniform closed balls conjecture", Control and Cybernetics 38 (2009) 1525--1534, and "Generalized exterior sphere conditions and φ-convexity", Discrete and Continuous Dynamical Systems, Serie B (to appear). We also provide a generalization of this conjecture to the variable radius case.
Keywords: Interior and exterior sphere conditions, union of uniform closed balls property, proximal analysis, nonsmooth analysis.
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