On the covering index of convex bodies

被引:6
作者
Bezdek, Karoly [1 ,2 ]
Khan, Muhammad A. [1 ]
机构
[1] Univ Calgary, Dept Math & Stat, Calgary, AB, Canada
[2] Univ Pannonia, Dept Math, Veszprem, Hungary
基金
加拿大自然科学与工程研究理事会;
关键词
Convex body; Hadwiger Covering Conjecture; Boltyanski-Hadwiger Illumination Conjecture; Covering index; Covering parameter; Illumination number; Illumination parameter; ILLUMINATION; BALL;
D O I
10.1007/s00010-016-0409-z
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce two relatives of the covering parameter called covering index and weak covering index, which upper bound well-studied quantities like the illumination number, the illumination parameter and the covering parameter of a convex body. Intuitively, the two indices measure how well a convex body can be covered by a relatively small number of homothets having the same relatively small homothety ratio. We show that the covering index is a lower semicontinuous functional on the Banach-Mazur space of convex bodies. We further show that the affine d-cubes minimize the covering index in any dimension d, while circular disks maximize it in the plane. Furthermore, the covering index satisfies a nice compatibility with the operations of direct vector sum and vector sum. In fact, we obtain an exact formula for the covering index of a direct vector sum of convex bodies that works in infinitely many instances. This together with a minimization property can be used to determine the covering index of infinitely many convex bodies. As the name suggests, the weak covering index loses some of the important properties of the covering index. Finally, we obtain upper bounds on the covering and weak covering index.
引用
收藏
页码:879 / 903
页数:25
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