On the Multiple Covering Densities of Triangles

TL;DR

This paper proves that for triangles, the minimum density of k-fold translative coverings equals that of lattice coverings, confirming a conjecture and extending understanding of covering densities in geometric arrangements.

Contribution

It establishes the equality of translative and lattice covering densities for triangles, building on recent results and confirming a conjecture in geometric covering theory.

Findings

quality of  and  covering densities for triangles

valuation of  covering density as  for all positive integers k

Extension of known results in geometric covering densities

Abstract

Given a convex disk $K$ and a positive integer $k$, let $\vartheta_T^k(K)$ and $\vartheta_L^k(K)$ denote the $k$-fold translative covering density and the $k$-fold lattice covering density of $K$, respectively. Let $T$ be a triangle. In a very recent paper, K. Sriamorn proved that $\vartheta_L^k(T)=\frac{2k+1}{2}$. In this paper, we will show that $\vartheta_T^k(T)=\vartheta_L^k(T)$.