Validity of the Union of Uniform Closed Balls Conjecture

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.