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