  The problem of covering a region of the plane with a fixed number of
minimum-radius identical balls is studied in the present work. An explicit
construction of bi-Lipschitz mappings is provided to model small perturbations
of the union of balls. This allows us to obtain analytical expressions for
first- and second-order derivatives using nonsmooth shape optimization
techniques under appropriate regularity assumptions. Singular cases are also
studied using asymptotic analysis. For the case of regions given by the union
of disjoint convex polygons, algorithms based on Voronoi diagrams that do not
rely on approximations are given to compute the derivatives. Extensive
numerical experiments illustrate the capabilities and limitations of the
introduced approach.
