Functionals on triangulations of Delaunay sets (Q2876650)

A Delone (=Delaunay) or \((R,r)\)-set in Euclidean space \(\mathbb E^{d}\) is a set with the properties that for each point of \(\mathbb E^{d}\) there is a point of the \((R,r)\)-set at distance at most \(R\) and any two points of the \((R,r)\)-set have distance greater than \(r\). Among the polytopal complexes in \(\mathbb E^{d}\) with a \((R,r)\)-set as set of vertices the \textit{Delone triangulation} is defined as the set of those convex polytopes where the circumsphere of the polytope contains no other points of the \((R,r)\)-set as its vertices (`empty sphere'). (That the Delone triangulation is a polytopal complex is a consequence of a theorem of \textit{P. M. Gruber} and \textit{S. S. Ryshkov} [Eur. J. Comb. 10, No. 1, 83--84 (1989; Zbl 0664.52011)]).NEWLINENEWLINEThe authors study the density of certain functionals of the polytopal complexes where the vertex sets are the points of a \((R,r)\)-set. They show that the density is minimum if the polytopal complex is the Delone triangulation, if this is true in the finite case.