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Alexey Glazyrin

Publications and source records attributed to Alexey Glazyrin.

25 records · Page 2Linked to original sources
arXiv2019

Energy on spheres and discreteness of minimizing measures

In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the $p$-frame energy has empty interior whenever $p$ is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.

math.CA↗
arXiv2016

Upper bounds for $s$-distance sets and equiangular lines

The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products $\{α, -α\}$, $α\in[0,1)$, are called equiangular. The problem of determining the maximum size of $s$-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an $s$-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in $\mathbb{R}^n$, $n\geq 7$, is $\frac{n(n+1)}2$ with possible exceptions for some $n=(2k+1)^2-3$, $k \in \mathbb{N}$. We also prove the universal upper bound $\sim \frac 2 3 n a^2$ for equiangular sets with $α=\frac 1 a$ and, employing this bound, prove a new upper bound on the size of equiangular sets in all dimensions. Finally, we classify all equiangular sets reaching this new bound.

math.MG↗
arXiv2014

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.

math.MG↗
arXiv2012

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 $(n+1)^{\frac {n-1} 2}$.

math.MG↗
arXiv2012

Functionals on Triangulations of Delaunay Sets

We study densities of functionals over uniformly bounded triangulations of a Delaunay set of vertices, and prove that the minimum is attained for the Delaunay triangulation if this is the case for finite sets.

math.MG↗
arXiv2011

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. In this paper, we show that there are certain gaps in Schatteman's proof, which is based on the Bruggesser-Mani shelling method. We show that using this 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.

math.MG↗