Related papers: Voronoi conjecture for five-dimensional paralleloh…
We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative…
We exploit the fact that two-dimensional facets of the Voronoi and Delone cells of the root lattice A_n in n-dimensional space are the identical rhombuses and equilateral triangles respectively.The prototiles obtained from orthogonal…
We show that the Voronoi cells of the lattice of integer flows of a finite connected graph $G$ in the quadratic vector space of real valued flows have the following very precise combinatorics: the face poset of a Voronoi cell is isomorphic…
Recently, Greenfeld and Tao disprove the conjecture that translational tilings of a single tile can always be periodic [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension…
Given a lattice $L$, a full dimensional polytope $P$ is called a {\em Delaunay polytope} if the set of its vertices is $S\cap L$ with $S$ being an {\em empty sphere} of the lattice. Extending our previous work \cite{DD-hyp} on the {\em…
2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level…
This article introduces the theory of Veronese polytopes, a broad generalisation of cyclic polytopes. These arise as convex hulls of points on curves with one or more connected components, obtained as the image of the rational normal curve…
In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results: Besides parallelograms and…
By a {\em Voronoi parallelotope} $P(a)$ we mean a parallelotope determined by a non-negative quadratic form $a$. It was studied by Voronoi in his famous memoir. For a set of vectors $\mathcal P$, we call its {\em dual} a set of vectors…
A polytope $D$ whose vertices belong to a lattice of rank $d$ is Delaunay if there is a circumscribing $d$-dimensional ellipsoid, $E$, with interior free of lattice points so that the vertices of $D$ lie on $E$. If in addition, the…
We give a proof in modern language of the following result by Paul Gordan and Max N\"other: a homogeneous quasi-translation in dimension $5$ without linear invariants would be linearly conjugate to another such quasi-translation $x + H$,…
Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that it has an edge-unfolding to a net [Gho14]. A net is a simple planar polygon; in particular, it does not self-overlap. One can view his result…
The Fuglede conjecture states that a set is spectral if and only if it tiles by translation. The conjecture was disproved by T. Tao for dimensions 5 and higher by giving a counterexample in $\mathbb{Z}_3^5$. We present a computer program…
The Hodge conjecture is shown to hold for rationally connected fivefolds, or more generally for fivefolds for which the base of the maximal rationally connected fibration is at most 3 dimensional.
Let $(\rho_\lambda\colon G_{\mathbb Q}\to \operatorname{GL}_5(\overline{E}_\lambda))_\lambda$ be a strictly compatible system of Galois representations such that no Hodge--Tate weight has multiplicity $5$. Under mild assumptions, we show…
Our main theoretical result is that, if a simple polytope has a pair of complementary vertices (i.e., two vertices with no facets in common), then it has at least two such pairs, which can be chosen to be disjoint. Using this result, we…
A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\Omega$ is a spectral set if and only if it can tile the…
In this work we prove constructively that the complement $\R^n\setminus\pol$ of a convex polyhedron $\pol\subset\R^n$ and the complement $\R^n\setminus\Int(\pol)$ of its interior are regular images of $\R^n$. If $\pol$ is moreover bounded,…
Generalizing results by Valette, Zamfirescu and Laczkovich, we will prove that a convex body $K$ is a polytope if there are sufficiently many tilings which contain a tile similar to $K$. Furthermore, we give an example that this can not be…
A numerically efficient, accurate, and easily implemented integration scheme over convex Voronoi polyhedra (VP) is presented for use in {\it ab-initio} electronic-structure calculations. We combine a weighted Voronoi tessellation with…