We prove that the complement of a closed set S satisfying an extended exterior sphere condition is nothing but the union of closed balls with common radius. This generalizes [11, Theorem 3] where the set S is assumed to ...

We consider sets S С R satisfying a certain exterior sphere condition, and it is shown that under wedgedness of S, it coincides with φ-convexity. We also offer related improvements concerning the union of uniform closed ...

For a given multifunction, we provide proximal differentiability condition under which the equivalence between the interior sphere condition of each value of the multifunction and the interior sphere condition of its graph holds.

We provide a new analytical proof for a strengthened version of the variable radius form of the union of closed balls conjecture. We also introduce a strong version of this conjecture and discuss its validity.

We generalize, to the variable radius case, the main results contained in a recent paper of the authors [A new class of sets regularity, J. Convex Analysis 25/4 (2018) 1059--1074], where a new regularity class, called ...