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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 2, pp. 313-324 (2001)
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Covering Belt Bodies by Smaller Homothetical Copies

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Vladimir Boltyanski; Horst Martini

CIMAT, A.P. 402, 36000 Guanajuato, Gto., Mexico
Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany

**Abstract:** \def\sR{** R}** Let $b(K)$ denote the minimal number of smaller homothetical copies of a convex body $K\subset\sR^n, n\ge 2$, covering $K$. For the class $\cal B$ of belt bodies, which is dense in the set of all convex bodies (in the Hausdorff metric), $3\cdot 2^{n-2}$ is known to be an upper bound on $b(K)$ if $K$ is different from a parallelotope. We will show that (except for all parallelotopes and two particular cases, each satisfying $b(K) = 3\cdot 2^{n-2}$) within $\cal B$ this bound can be improved to $5\cdot 2^{n-3}$.

**Keywords:** belt body, belt polytope, Hadwiger's covering problem, homothetical copy, (outer) illumination, zonoid, zonotope

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