Bounds for totally separable translative packings in the plane

TL;DR

This paper establishes new bounds and inequalities for totally separable translative packings of convex domains in the plane, including density, convex hull area, and covering ratio, advancing understanding of their geometric properties.

Contribution

It proves an analogue of Oler's inequality for these packings and derives new bounds on density, convex hull size, and covering ratios for totally separable packings.

Findings

Largest density of totally separable translative packings determined.

Smallest convex hull area for packings with given translates found.

Maximum covering ratio of soft disk packings established.

Abstract

A packing of translates of a convex domain in the Euclidean plane is said to be totally separable if any two packing elements can be separated by a line disjoint from the interior of every packing element. This notion was introduced by G. Fejes T\'oth and L. Fejes T\'oth (1973) and has attracted significant attention. In this paper we prove an analogue of Oler's inequality for totally separable translative packings of convex domains and then we derive from it some new results. This includes finding the largest density of totally separable translative packings of an arbitrary convex domain and finding the smallest area convex hull of totally separable packings (resp., totally separable soft packings) generated by given number of translates of a convex domain (resp., soft convex domain). Finally, we determine the largest covering ratio (that is, the largest fraction of the plane covered by the soft disks) of an arbitrary totally separable soft disk packing with given soft parameter.