On the Covering Densities of Quarter-Convex Disks

TL;DR

This paper proves a conjecture relating translative and lattice covering densities for a class of convex disks called quarter-convex disks, extending known results from symmetric disks and triangles.

Contribution

It establishes the conjecture for quarter-convex disks, a broad class including triangles and quadrilaterals, advancing understanding of covering densities.

Findings

Proved the conjecture for quarter-convex disks

Extended results from symmetric disks and triangles

Includes all convex quadrilaterals

Abstract

It is conjectured that for every convex disks K, the translative covering density of K and the lattice covering density of K are identical. It is well known that this conjecture is true for every centrally symmetric convex disks. For the non-symmetric case, we only know that the conjecture is true for triangles. In this paper, we prove the conjecture for a class of convex disks (quarter-convex disks), which includes all triangles and convex quadrilaterals.