The Covering Index of Convex Bodies

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 [Mathematika 52 (2005), 47–52] introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce a relative of the covering parameter called covering index, which turns out to have a number of nice properties. Intuitively, the covering index measures 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 and provides a useful upper bound for well-studied quantities like the illumination number, the illumination parameter, the vertex index and the covering parameter of a convex body. We obtain upper bounds on the covering index and investigate its optimizers. We show that the affine d-cubes minimize covering index in any dimension d, while circular disks maximize it in the plane. Furthermore, we show that 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 monotonicity property can be used to determine the covering index of infinitely many convex bodies.