Discrete & Computational Geometry

, Volume 53, Issue 2, pp 245–260 | Cite as

The Voronoi Conjecture for Parallelohedra with Simply Connected \(\delta \)-Surfaces



We show that the Voronoi conjecture is true for parallelohedra with simply connected \(\delta \)-surfaces. That is, we show that if the boundary of parallelohedron \(P\) remains simply connected after removing closed nonprimitive faces of codimension 2, then \(P\) is affinely equivalent to a Dirichlet–Voronoi domain of some lattice. Also, we construct the \(\pi \)-surface associated with a parallelohedron and give another condition in terms of a homology group of the constructed surface. Every parallelohedron with a simply connected \(\delta \)-surface also satisfies the condition on the homology group of the \(\pi \)-surface.


Parallelohedron Voronoi conjecture Fundamental group Homology group Canonical scaling 



The authors would like to thank Professor Nikolai Dolbilin and Professor Robert Erdahl for fruitful discussions and helpful comments and remarks. The first author is supported by RScF project 14-11-00414.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at BrownsvilleBrownsvilleUSA
  2. 2.Higher School of Economics, Faculty of Computer ScienceMoscowRussia
  3. 3.Center for Optical Neural TechnologiesScientific Research Institute for Systems Analysis, RASMoscowRussia
  4. 4.Steklov Mathematical InstituteMoscowRussia

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