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Covering a three-dimensional convex body by smaller homothetic copies. (English) Zbl 0912.52012

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 \(\frac{7}{8}\).

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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