Structural Packing and Covering Problems

Abstract

Let k be a natural number and $D \subseteq \mathbb{R}^d$ an arbitrary domain. An arrangement $\mathcal{C}$ of convex bodies in ℝ^d is said to form a k-fold covering of D if every point of D belongs to at least k members of $\mathcal{C}$, and it is called a k-fold packing if every point belongs to the interior of at most k members of $\mathcal{C}$. Thus, the usual coverings and packings are the same as the 1-fold (or simple) coverings and packings, respectively. The survey paper of G. Fejes Tóth [FeT83] gives an almost complete account of the known results about the thinnest k-fold coverings and densest k-fold packings of ℝ^d with congruent copies of a convex body $\mathcal{C}$. These questions are usually quite difficult, and they often require somewhat technical extensions of the methods developed for the case k = 1 (see, e.g., Few [Fe64], [Fe67], [Ga96], [Bl99].) G. Fejes Tóth [FeT76], [FeT77], [FeT79] generalized the concept of Voronoi-Dirichlet cell decomposition to show that the densities δ ^k(B ²) and θ ^k(B ²) of the densest k-fold packing and thinnest k-fold covering of the plane with unit circles (disks) satisfy

$$ \begin{gathered} \delta ^k (B^2 ) \leqslant \frac{\pi } {6}\cot \frac{\pi } {{6k}} < k - \frac{{\pi ^2 }} {{108k}}, \hfill \\ \theta ^k (B^2 ) \geqslant \frac{\pi } {3}\csc \frac{\pi } {{3k}} > k + \frac{{\pi ^2 }} {{54k}}, \hfill \\ \end{gathered} $$

for every k. Improving some earlier results of Cohn [Co76] and Groemer [Gr86], Bolle [Bo84], [Bo89] and Huxley [Hu93] showed that for the corresponding densities δ ^k _L (B ²) and θ ^k _L (B ²), restricted to lattice packings and coverings, respectively, we have

$$ \begin{gathered} k - ck^{\tfrac{{23}} {{73 + \varepsilon }}} \leqslant \delta _L^k (B^2 ) \leqslant k - c'k^{\tfrac{1} {4}} , \hfill \\ k + c'k^{\tfrac{1} {4}} \leqslant \theta _L^k (B^2 ) \leqslant k + ck^{\tfrac{{23}} {{73 + \varepsilon }}} , \hfill \\ \end{gathered} $$

for any ε > 0 and for suitable c, c′ > 0 depending only on ε.