Voronoi tilings, toric arrangements and degenerations of line bundles I

高秩内积、Voronoi镶嵌与环面的度量退化

Higher rank inner products, Voronoi tilings and metric degenerations of tori

该研究在实与复向量空间上引入高秩内积并研究对应Voronoi镶嵌，还借助该框架刻画极化环面的度量退化及离散集Voronoi镶嵌的Hausdorff极限。

Voronoi tilings hidden in crystals - The case of maximal abelian coverings

Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadri as a by-product of Geometric Invariant Theory, he shows that the Voronoi tiling (also known as the Wigner-Seitz tiling) is hidden in the crystal, that is, the crystal does not intrude the interiors of the top-dimensional Voronoi cells. The result turns out to be closely related…

Voronoi tilings, toric arrangements and degenerations of line bundles II

We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This article and its first and third part companion parts are the first in a series aimed to explore this new approach. In the first part, we set up the combinatorial framework and showed how graphs weighted with integer lengths associated to the edges provide tilings of Euclidean spaces by polytope…

The second Voronoi conjecture on parallelohedra for zonotopes

We prove the second Voronoi conjecture on parallelohedra for zonotope. We show that for a given face-to-face tiling of d-dimensional Euclidean space into parallel copies of zonotope Z there are d vectors, connecting centers of zonotopes with common facet, that are basis of the correspondent lattice of the tiling.

Once more about Voronoi's conjecture and space tiling zonotopes

Voronoi conjectured that any parallelotope is affinely equivalent to a Voronoi polytope. A parallelotope is defined by a set of $m$ facet vectors $p_i$ and defines a set of $m$ lattice vectors $t_i$, $1\le i\le m$. We show that Voronoi's conjecture is true for an $n$-dimensional parallelotope $P$ if and only if there exist scalars $γ_i$ and a positive definite $n\times n$ matrix $Q$ such that $γ_i p_i=Qt_i$ for all $i$. In this case the quadratic form $f(x)=x^TQx$ is the metric form of $P$. As an example, we consider in detail the case of a zonotopal parallelotope. We show that $Q=(Z_βZ^T_β)^{…

A Note on Space Tiling Zonotopes

In 1908 Voronoi conjectured that every convex polytope which tiles space face-to-face by translations is affinely equivalent to the Dirichlet-Voronoi polytope of some lattice. In 1999 Erdahl proved this conjecture for the special case of zonotopes. A zonotope is a projection of a regular cube under some affine transformation. In 1975 McMullen showed several equivalent conditions for a zonotope to be a space tiling zonotope, i.e. a zonotope which admits a face-to-face tiling of space by translations. Implicitly, he related space tiling zonotopes to a special class of oriented matroids (regular …