Coating by cubes
Consider \(P_ 0,P_ 1,\dots,P_ n\) convex \(d\)-polytopes in \(\mathbb{E}^ d\) \((d \geq 2)\) with pairwise disjoint interiors. Then \(P_ 0\) is said to be coated by \(P_ 1,\dots,P_ n\) if \(P_ 0 \subset \text{int}(\bigcup^ n_{i = 0} P_ i)\), where \(\text{int}()\) denotes the interior of a set. The authors are interested in the minimal number of convex polytopes that coat a given convex polytope. More precisely they investigate the following two problems: \(\bullet\) What is the minimum number of \(d\)-simplices that can coat a \(d\)-simplex in \(\mathbb{E}^ d\)? \(\bullet\) Is \(2^{d - 1} + 2\) the minimum number of \(d\)-cubes that can coat a \(d\)-cube? Using elementary methods they prove the following two results in this direction: \(\bullet\) Every \(d\)-simplex can be coated by \((2d - 1)\) \(d\)-simplices in \(\mathbb{E}^ d\). \(\bullet\) The minimum number of translates of a \(d\)-cube that can coat a given \(d\)-cube in \(\mathbb{E}^ d\) is at least \(2^ d\). If all \(d\)-cubes are translates of each other, then \(2^ d\) can be replaced by \(2^{d + 1} - 2\).
scientific article; zbMATH DE number 702462