Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 2, pp. 313-324 (2001)

Covering Belt Bodies by Smaller Homothetical Copies

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

Full text of the article:

[Previous Article] [Next Article] [Contents of this Number]
© 2001 ELibM for the EMIS Electronic Edition