On the union of closed balls property
The authors propose in this paper a new analytical proof for a strengthened version of the variable radius form of the union of closed balls conjecture introduced by the same authors in an old paper. The improvement results from the fact that they provide a bigger radius for the balls covering the given closed set. The main keys of the proof are the new analytical characterizations of the \(\psi\)-union of closed balls property and the \(\theta\)-exterior sphere condition together with some techniques from nonsmooth analysis. As a corollary, the authors deduce a new analytical proof for the weak version of the union of closed balls conjecture. They also introduce a strong version of the variable radius form of this conjecture and prove, by a counterexample, that the geometric methods used in the old paper to deduce the variable radius form from the constant radius one, do not work in the strong case.
scientific article; zbMATH DE number 6136278
scientific article