Alexey Glazyrin

Lower bounds for the simplexity of the n -cube

In this paper we prove a new asymptotic lower bound for the minimal number of simplices in simplicial dissections of n -dimensional cubes. In particular we show that the number of simplices in dissections of n -cubes without additional vertices is at least ( n + 1 ) n - 1 2 .

Stability of the Simplex Bound for Packings by Equal Spherical Caps Determined by Simplicial Regular Polytopes

It is well known that the vertices of any Euclidean simplicial regular polytope determine an optimal packing of equal spherical balls. We prove a stability version of optimal order of this result.

On simplicial partitions of polytopes

We prove some general properties of prismoids, i.e., polytopes all of whose vertices lie in two parallel planes. On the basis of these properties, we obtain a nontrivial lower bound for the number of simplices in a triangulation of the n-dimensional cube.

The Voronoi functional is maximized by the Delaunay triangulation in the plane

We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.

The extremal spheres theorem

Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well known that for any convex polygon there exist at least two empty and at least two full circles, i.e. at least four extremal circles. In 1990 Schatteman considered a generalization of this theorem for convex polytopes in d-dimensional Euclidean space. Namely, he claimed that there exist at least 2d extremal neighboring spheres for generic polytopes. His proof is based on the Bruggesser-Mani shelling method. In this paper, we show that there are certain gaps in Schattemans proof. We also show that using the Bruggesser-Mani-Schatteman method it is possible to prove that there are at least d+1 extremal neighboring spheres. However, the existence problem of 2d extremal neighboring spheres is still open.