On the Structure of Higher Order Voronoi Cells

The classic Voronoi cells can be generalized to a higher-order version by considering the cells of points for which a given $k$-element subset of the set of sites consists of the $k$ closest sites. We study the structure of the $k$-order Voronoi cells and illustrate our theoretical findings with a case study of two-dimensional higher-order Voronoi cells for four points.Comment: Minor correction

On the structure of higher order Voronoi cells

The classic Voronoi cells can be generalized to a higher order version by considering the cells of points for which a given k-element subset of the set of sites consists of the k closest sites. We study the structure of the k-order Voronoi cells and illustrate our theoretical findings with a case study of two-dimensional higher order Voronoi cells for four points

On location-allocation problems for dimensional facilities

This paper deals with a bilevel approach of the location-allocation problem with dimensional facilities. We present a general model that allows us to consider very general shapes of domains for the dimensional facilities and we prove the existence of optimal solutions under mild, natural assumptions. To achieve these results we borrow tools from optimal transport mass theory that allow us to give explicit solution structure of the considered lower level problem. We also provide a discretization approach that can approximate, up to any degree of accuracy, the optimal solution of the original problem. This discrete approximation can be optimally solved via a mixedinteger linear program. To address very large instance sizes we also provide a GRASP heuristic that performs rather well according to our experimental results. The paper also reports some experiments run on test data.Ministerio de Economía y Competitividad (MINECO). Españ

The geometry of optimal partitions in location problems

Given the position of some facilities, we study the shape of optimal par- titions of the customers’ area in a general planar demand region minimizing total average cost that depends on a set up cost plus some function of the travelling dis- tances. By taking into account different norms, according to the considered situation of the location problem, we characterize optimal consumers’ partitions and describe their geometry. The case of dimensional facilities is also investigated