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Properties of faces of parallelohedra

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Abstract

We consider an important class of polytopes, called parallelohedra, that tile the Euclidean space. The concepts of a standard face of a parallelohedron and of the index of a face are introduced. It is shown that the sum of indices of standard faces in a parallelohedron is an invariant; this implies the Minkowski bound for the number of facets of parallelohedra. New properties of faces of parallelohedra are obtained.

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Authors and Affiliations

  1. Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    N. P. Dolbilin

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  1. N. P. Dolbilin
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Correspondence to N. P. Dolbilin.

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Original Russian Text © N.P. Dolbilin, 2009, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 266, pp. 112–126.

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Dolbilin, N.P. Properties of faces of parallelohedra. Proc. Steklov Inst. Math. 266, 105–119 (2009). https://doi.org/10.1134/S0081543809030067

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