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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

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 \(f_ n(C)=2^ n-1\).

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
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