Bezdek, András; Bezdek, Károly A sufficient condition for the covering of the unit cube by homothetic copies in the \(n\)-dimensional Euclidean space. (Eine hinreichende Bedingung für die Überdeckung des Einheitswürfels durch homothetische Exemplare im \(n\)-dimensionalen euklidischen Raum.) (German) Zbl 0556.52008 Beitr. Algebra Geom. 17, 5-21 (1984). L. Fejes Tóth, independently of D. J. Newman [SIAM Rev. 24, 77 (1982),http://dx.doi.org/10.1137/1024007] phrased the following problem: Let \(K\) be a convex body of volume 1 in the \(n\)-dimensional Euclidean space \(E^ n\). Denote by \(f_ n(K)\) the smallest positive number which has the following property. Any collection of bodies, which are homothetic to \(K\) with total volume \(f_ n(K)\) can cover \(K\). The problem is to determine \(f_ n(K)\). L. Fejes Tóth conjectured that \(2\leq f_ 2(K)\leq 3\) and \(f_ n(C)=2^ n-1,\) where \(C\) denotes the unit hypercube in \(E^ n\). The paper contains the proofs of the statements \(f_ n(C)=2^ n-1,\) \(2\leq f_ 2(K)<12\) and discusses some remarks about similar questions. Independently of the authors A. Meir [SIAM Rev. 25, No. 1, 99–101 (1983), http://dx.doi.org/10.1137/1025012] published a proof for \(f_ n(C)=2^ n-1\). Cited in 1 ReviewCited in 5 Documents MSC: 52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry) 52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces) 52A40 Inequalities and extremum problems involving convexity in convex geometry Keywords:homothetic covering; hypercube PDFBibTeX XMLCite \textit{A. Bezdek} and \textit{K. Bezdek}, Beitr. Algebra Geom. 17, 5--21 (1984; Zbl 0556.52008) Full Text: EuDML