The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms

Abstract

Given a set of constrained vertex norms, we proved the existence of a convex configuration with respect to the set of distinct constrained vertex norms in the two-dimensional case when the constrained vertex norms are distinct or repeated for, at most, four points. However, we proved that there always exists a convex configuration in the three-dimensional case. In the application, we can imply the existence of the non-empty spherical Laguerre Voronoi diagram.

Keywords

• convex polyhedron
• convex configuration
• spherical Laguerre Voronoi diagram