{"title":"On Combinatorics of Voronoi Polytopes for Perturbations of the Dual Root Lattices","authors":"A. Garber","doi":"10.1080/10586458.2021.1994488","DOIUrl":null,"url":null,"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.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Experimental Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/10586458.2021.1994488","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
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.
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