Existence of a Convex Polyhedron with Respect to the Given Radii

Given a set of radii measured from a fixed point, the existence of a convex configuration with respect to the set of distinct radii in the two-dimensional case is proved when radii are distinct or repeated 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.

[1]  Günter Rote,et al.  Blowing Up Polygonal Linkages , 2003 .

[2]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[3]  Thomas Ottmann,et al.  Enumerating Extreme Points in Higher Dimensions , 2001, Nord. J. Comput..

[4]  Bahman Kalantari,et al.  A characterization theorem and an algorithm for a convex hull problem , 2012, Ann. Oper. Res..

[5]  Monique Teillaud,et al.  Robust and Efficient Delaunay Triangulations of Points on Or Close to a Sphere , 2010, SEA.

[6]  F. P. Preparata,et al.  Convex hulls of finite sets of points in two and three dimensions , 1977, CACM.

[7]  Sue Whitesides,et al.  Reconfiguring closed polygonal chains in Euclideand-space , 1995, Discret. Comput. Geom..

[8]  J. Gallier Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations , 2008, 0805.0292.

[9]  Kenneth L. Clarkson,et al.  More output-sensitive geometric algorithms , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[10]  Erik D. Demaine,et al.  Reconfiguring convex polygons , 2001, Comput. Geom..

[11]  Hiroshi Imai,et al.  Voronoi Diagram in the Laguerre Geometry and its Applications , 1985, SIAM J. Comput..

[12]  J. Dulá,et al.  A new procedure for identifying the frame of the convex hull of a finite collection of points in multidimensional space , 1996 .

[13]  Kokichi Sugihara Three-dimensional convex hull as a fruitful source of diagrams , 2000, Theor. Comput. Sci..

[14]  K. Sugihara Laguerre Voronoi Diagram on the Sphere , 2002 .