On the density of the thinnest covering of \(\mathbb{R}^n\)
Let \(K\) be a convex body in \(\mathbb{R}^n\). Let \(\mathcal{U}(K)\) denote the family of all coverings \(\{K + x_i\}_{i\geq 1}\) of the entire space \(\mathbb{R}^n\), where \(x_i\in \mathbb{R}^n\), \(i\geq 1\). Given \(s>0\), let \(Q_s = \prod _{i\leq n} [-s, s)\) and \(R_s = \prod _{i\leq n} (-s, s)\). The covering density of \(K\) is defined as \[ \theta (K) = \inf_{\{K + x_i\}_{i}\in \mathcal{U}(K)} \liminf _{s\to \infty} \frac{1}{\mbox{vol} (Q_s)} \sum_{K+x_i \subset Q_s } \mbox{vol}(K+x_i). \] It is known that \[ \theta (K) = \inf_{\{K + x_i\}_{i}\in \mathcal{U}(K)} \liminf _{s\to \infty} \frac{1}{\mbox{vol} (Q_s)} \sum_{(K+x_i)\cap Q_s = \emptyset } \mbox{vol} (K+x_i). \] It follows from results of \textit{H.~Groemer} [Math. Z. 81, 260--278 (1963; Zbl 0123.39104)] that for any fixed \(t>0\), \begin{align*} \theta (K) &= \lim_{s\to \infty }\inf_{\{K + x_i\}_{i}\in \mathcal{U}(K)} \frac{1}{\mbox{vol} (Q_s)} \sum_{(K+x_i)\cap Q_s = \emptyset } \mbox{vol} (K+x_i) \\ &= \lim_{\lambda \to 0^+ }\inf_{\{\lambda K + x_i\}_{i}\in \mathcal{U}(\lambda K)} \frac{1}{\mbox{vol} (R_t)} \sum_{(\lambda K+x_i)\cap R_t = \emptyset } \mbox{vol}(\lambda K+x_i) . \end{align*} The authors provide another proof of this result. They also mention that yet another proof given in [\textit{S.-M. Jung}, Commun. Korean Math. Soc. 10, No. 3, 621--632 (1995, Zbl 0943.52002)] is based on an unverified assertion and contains some ambiguities.
scientific article; zbMATH DE number 7229053
scientific article