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Minkowski bisectors, Minkowski cells and lattice coverings

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Abstract

The Dirichlet–Voronoi cell and parallelohedron are fundamental concepts in Geometry. In particular, they do play important roles in the study of ball packing and ball covering. However, to study packing and covering of general convex bodies, they are no longer so useful (see Theorem 0). By introducing Minkowski bisectors and Minkowski cells, this paper explores a new way to study the density \(\theta ^*(C)\) of the thinnest lattice covering of \(\mathbb {E}^n\) by a centrally symmetric convex body C. Several basic results (Theorems 2 and 4, Corollary 1) and unexpected geometric phenomena (Theorem 0, Example 1, Remark 4) about Minkowski bisectors, Minkowski cells and covering densities are discovered.

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Acknowledgements

For some useful comments and revision suggestions, we are grateful to Prof. S. Wu and the referees. This work is supported by 973 Programs 2013CB834201 and 2011CB302401, and the Chang Jiang Scholars Program of China.

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  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    C. Zong

  2. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    F. Xue & C. Zong

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  1. F. Xue
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  2. C. Zong
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Correspondence to C. Zong.

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Xue, F., Zong, C. Minkowski bisectors, Minkowski cells and lattice coverings. Geom Dedicata 188, 123–139 (2017). https://doi.org/10.1007/s10711-016-0208-7

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