187 citations · 232 across the 22 of their papers we have counts for
16 papers · 1 filter
On the Voronoi Conjecture for combinatorially Voronoi parallelohedra in dimension five
Mathieu Dutour Sikirić, Alexey Garber, Alexander Magazinov
In a recent paper Garber, Gavrilyuk and Magazinov proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for…
On a conjecture by Eckhoff and Dolnikov concerning line transversals to Euclidean disks
Alexander Magazinov
Let be a convex body in the Euclidean plane . We say that a point set satsfies the property if the family of translates $\{ K + x…
A proof of a conjecture by Haviv, Lyubashevsky and Regev on the second moment of a lattice Voronoi cell
Alexander Magazinov
In this short note we prove a sharp lower bound for the second moment of a lattice Voronoi cell in terms of the respective covering radius. This gives an affirmative answer to a co…
A center transversal theorem for an improved Rado depth
Pavle V. M. Blagojević, Roman Karasev, Alexander Magazinov
A celebrated result of Dol'nikov, and of Živaljević and Vrećica, asserts that for every collection of measures on the Euclidean space th…
An improvement on the Rado bound for the centerline depth
Alexander Magazinov, Attila Pór
Let be a Borel probability measure in . For a -flat consider the value , where runs through all half-spaces containing . This infimum is c…
The sign-sequence constant of the plane
Ben Lund, Alexander Magazinov
Let be a finite-dimensional real normed space, and let be the unit ball in . The sign sequence constant of is the least such that, for each sequence $v_1, \ldo…
On Delaunay's classification theorem on faces of parallelohedra of codimension three
Alexander Magazinov
In 1929 B.~N.~Delaunay proved that there are exactly 5 types of coincidence of parallelohedra at faces of codimension 3. We give a combinatorial proof of this theorem and prove sev…
Fair partitioning by straight lines
Augustin Fruchard, Alexander Magazinov
A pizza is a pair of planar convex bodies ,where represents the dough and the topping of the pizza. A partition of a pizza by straight lines is a succession o…
Positive-fraction intersection results and variations of weak epsilon-nets
Alexander Magazinov, Pablo Soberón
Given a finite set of points in and a family of sets generated by the pairs of points of , we determine volumetric and structural conditions for the sets that allo…
Perfect prismatoids are lattice Delaunay polytopes
Marina Kozachok, Alexander Magazinov
A perfect prismatoid is a convex polytope such that for every its facet the set belongs to a supporting hyperplane . We prove that…
On a problem by Dol'nikov
Jesús Jerónimo-Castro, Alexander Magazinov, Pablo Soberón
In 2011 at an Oberwolfach workshop in Discrete Geometry, V. Dol'nikov posed the following problem. Consider three non-empty families of translates of a convex compact set in th…
A criterion of reducibility for a parallelohedron
Andrei Ordine, Alexander Magazinov
A parallelohedron is called reducible, if it can be represented as a direct product of two parallelohedra of lower dimension. In his Ph.D. thesis (2005) the first author proved a c…
The Voronoi conjecture for parallelohedra with simply connected -surface
Alexey Garber, Andrey Gavrilyuk, Alexander Magazinov
We show that the Voronoi conjecture is true for parallelohedra with simply connected -surface. Namely, we show that if the boundary of parallelohedron remains simply connect…
On the sum of the Voronoi polytope of a lattice with a zonotope
Mathieu Dutour Sikiric, Viatcheslav Grishukhin, Alexander Magazinov
A parallelotope is a polytope that admits a facet-to-facet tiling of space by translation copies of along a lattice. The Voronoi cell of a lattice is an exampl…
An upper bound for a valence of a face in a parallelohedral tiling
Alexander Magazinov
Consider a face-to-face parallelohedral tiling of and a -dimensional face of the tiling. We prove that the valence of (i.e. the number of tiles contain…
Asymptotics for some combinatorial characteristics of the convex hull of a Poisson point process in the Clifford torus
Alexander Magazinov
N. Dolbilin and M. Tanemura studied the convex hulls of finite subsets of the Clifford torus in . They have completely studied the combinatorial structure of the convex hu…