Lassak, Marek Covering a three-dimensional convex body by smaller homothetic copies. (English) Zbl 0912.52012 Beitr. Algebra Geom. 39, No. 2, 259-262 (1998). For \(C \subset E^d\) denote by \(L(C)\) the smallest \(m\) such that \(C\) can be covered by \(m\) its homothetic copies of a positive ratio smaller than 1. The well known Hadwiger’s conjecture says that \(L(C) \leq 2^d\) and is open for \(d \geq 3\).In the paper under review, the author proves that \(L(C) \leq 20\) for \(d=3\). In this case the homothety ratio depends on \(C\). It is also shown that \(L(C) \leq 24\) with a universal ratio \(\frac{7}{12}\sqrt{2}+\frac{1}{6}=0.9916\dots\) and \(L(C) \leq 28\) with a universal ratio \(\frac{7}{8}\). Reviewer: Peter Boyvalenkov (Sofia) Cited in 1 Document MSC: 52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry) Keywords:covering; Hadwiger’s conjecture PDFBibTeX XMLCite \textit{M. Lassak}, Beitr. Algebra Geom. 39, No. 2, 259--262 (1998; Zbl 0912.52012) Full Text: EuDML