Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 2, pp. 411-416 (2000)

On Rhombic Dodecahedra

Karoly Bezdek

Eötvös University, Institute of Mathematics II, Department of Geometry, H-1088 Budapest, Rakoczi ut 5, Hungary, e-mail: and Cornell University, Department of Mathematics, Malott Hall, Ithaca, NY 14853-7901, USA, e-mail:

Abstract: In this note we prove that the intrinsic $i$-volume of any $d$-dimensional zonotope generated by $d+1$ (resp. $d$) line segments and containing a $d$-dimensional unit ball in $\bf{E}^d$ is at least as large as the intrinsic $i$-volume of the $d$-dimensional regular zonotope generated by $d+1$ line segments having inradius 1, where $i=1,\dots,d-1,d$.

Keywords: zonotopes, parallelohedra, rhombic dodecahedra, lattice sphere packings

Classification (MSC2000): 52C17, 52A40, 52B60, 52A38

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