16 citations · 47 across the 18 of their papers we have counts for
22 papers
New constructions of optimal arrangements of lines in
Alexey Glazyrin
In this paper we provide new constructions of equiangular tight frames of size in . We generalize the doubling construction of Fallon and Iverson to a tensor mul…
On the total perimeter of disjoint convex bodies
Arseniy Akopyan, Alexey Glazyrin
In this note we introduce a pseudometric on convex planar curves based on distances between normal lines and show its basic properties. Then we use this pseudometric to give a shor…
Optimal Measures for Multivariate Geometric Potentials
Dmitriy Bilyk, Damir Ferizović, Alexey Glazyrin +3
We study measures and point configurations optimizing energies based on multivariate potentials. The emphasis is put on potentials defined by geometric characteristics of sets of p…
Optimizers of three-point energies and nearly orthogonal sets
Dmitriy Bilyk, Damir Ferizović, Alexey Glazyrin +3
This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of…
Extensions of polynomial plank covering theorems
Alexey Glazyrin, Roman Karasev, Alexandr Polyanskii
We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to…
On the size of maximal binary codes with 2, 3, and 4 distances
Alexander Barg, Alexey Glazyrin, Wei-Jiun Kao +3
We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the e…
Covering by planks and avoiding zeros of polynomials
Alexey Glazyrin, Roman Karasev, Alexandr Polyanskii
We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex…
Potential theory with multivariate kernels
Dmitriy Bilyk, Damir Ferizović, Alexey Glazyrin +3
In the present paper we develop the theory of minimization for energies with multivariate kernels, i.e. energies, in which pairwise interactions are replaced by interactions betwee…
A short solution of the kissing number problem in dimension three
Alexey Glazyrin
In this note, we give a short solution of the kissing number problem in dimension three.
Domes over curves
Alexey Glazyrin, Igor Pak
A closed piecewise linear curve is called integral if it is comprised of unit intervals. Kenyon's problem asks whether for every integral curve in , there is a do…
Energy on spheres and discreteness of minimizing measures
Dmitriy Bilyk, Alexey Glazyrin, Ryan Matzke +2
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 meas…
Optimal measures for p-frame energies on spheres
Dmitriy Bilyk, Alexey Glazyrin, Ryan Matzke +2
We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the -frame energies, i.e. energie…
Covering by homothets and illuminating convex bodies
Alexey Glazyrin
The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number and a convex body , is the infimum of …
Moments of isotropic measures and optimal projective codes
Alexey Glazyrin
In this paper, we use the linear programming approach to find new upper bounds for the moments of isotropic measures. These bounds are then utilized for finding lower packing bound…
Repeated minimizers of -frame energies
Alexey Glazyrin, Josiah Park
For a collection of unit vectors , define the -frame energy of as the quantity . In this…
Stability of optimal spherical codes
Károly J. Böröczky, Alexey Glazyrin
For many extremal configurations of points on a sphere, the linear programming approach can be used to show their optimality. In this paper we establish the general framework for s…
Stability of the simplex bound for packings by equal spherical caps determined by simplicial regular polytopes
Károly Böröczky, Károly J. Böröczky, Alexey Glazyrin +1
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…
Upper bounds for -distance sets and equiangular lines
Alexey Glazyrin, Wei-Hsuan Yu
The set of points in a metric space is called an -distance set if pairwise distances between these points admit only distinct values. Two-distance spherical sets with the se…
The Voronoi Functional is Maximized by the Delaunay Triangulation in the Plane
Herbert Edelsbrunner, Alexey Glazyrin, Oleg R. Musin +1
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 Dela…
Functionals on Triangulations of Delaunay Sets
Nikolay P. Dolbilin, Herbert Edelsbrunner, Alexey Glazyrin +1
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…
Lower bounds for the simplexity of the n-cube
Alexey Glazyrin
In this paper we prove a new asymptotic lower bound for the minimal number of simplices in simplicial dissections of -dimensional cubes. In particular we show that the number of…
The Extremal Spheres Theorem
Arseniy Akopyan, Alexey Glazyrin, Oleg R. Musin +1
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 o…