Generalization of an interpolant using Voronoi diagrams in two directions

Recently the authors found a local coordinate property based on the planar Voronoi diagram that is simpler than the famous Sibson's local coordinates (R. Sibson, 1980; 1981), and proposed an interpolant using this property. The paper generalizes this property to general dimensions. The proof given in the paper enables us to use more general Voronoi diagrams, e.g., Laguerre Voronoi diagrams and numerically distributed Voronoi diagrams obtained by topology oriented algorithms. The paper also generalizes the author's interpolant to continuously distributed data sites.