On the Density of the Thinnest Covering of \(\mathbb{R}^n\)

Abstract

Let \(K\) be a compact convex subset of \(\mathbb{R}^n\) whose Lebesgue measure is positive and which includes the origin of the coordinate system. In this paper, the density of the thinnest covering of \(\mathbb{R}^n\) by translates of \(K\) will be transformed into another form that seems easier to use. Finally, in the discussion section of this paper, we will look at the applications of the main results.