On Combinatorics of Voronoi Polytopes for Perturbations of the Dual Root Lattices

IF 0.7 4区数学 Q2 MATHEMATICS

Experimental Mathematics Pub Date : 2021-04-16 DOI:10.1080/10586458.2021.1994488

A. Garber

引用次数: 1

Abstract

The Voronoi conjecture on parallelohedra claims that for every convex polytope P that tiles Euclidean d-dimensional space with translations there exists a d-dimensional lattice such that P and the Voronoi polytope of this lattice are affinely equivalent. The Voronoi conjecture is still open for the general case but it is known that some combinatorial restriction for the face structure of P ensure that the Voronoi conjecture holds for P . In this paper we prove that if P is the Voronoi polytope of one of the dual root lattices D∗ d , E∗ 6 , E∗ 7 or E∗ 8 = E8 or their small perturbations, then every parallelohedron combinatorially equivalent to P in strong sense satisfies the Voronoi conjecture.

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对偶根格摄动的Voronoi多面体组合

平行四边形上的Voronoi猜想声称，对于每一个用平移划分欧几里得d维空间的凸多面体P，都存在一个d维格，使得P和这个格的Vorononi多面体是仿射等价的。Voronoi猜想对一般情况仍然是开放的，但已知对P的面结构的一些组合限制确保了Voronoii猜想对P成立。本文证明了如果P是对偶根格D*D、E*6、E*7或E*8=E8或它们的小扰动之一的Voronoi多面体，则在强意义上与P组合等价的每个平行多面体都满足Voronoi猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.