Minkowski bisectors, Minkowski cells and lattice coverings

作者：Xue, F.; Zong, C.^*

来源：Geometriae Dedicata, 2017, 188(1): 123-139.

DOI：10.1007/s10711-016-0208-7

• Lattice
• Bisector
• Dirichlet-Voronoi cell
• Minkowski metric
• Minkowski bisector
• Minkowski cell
• Lattice covering

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 of the thinnest lattice covering of 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.

• 出版日期2017-6
• 单位北京大学; 天津大学