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→Known results: Clarified the term "symmetric" and emphasized that 3-dimensional applies to the whole sentence. Tags: Mobile edit Mobile web edit |
→Known results: Corrected an otherwise wrong statement. Tags: Mobile edit Mobile web edit |
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where <math>c</math> is a positive constant. For small <math>n</math> the upper bound of <math>(n+1)n^{n-1}-(n-1)(n-2)^{n-1}</math> established by {{harvtxt|Lassak|1988}} is better than the asymptotic one. In three dimensions it is known that 16 copies always suffice, but this is still far from the conjectured bound of 8 copies.<ref name="BMP">{{harvtxt|Brass|Moser|Pach|2005}}.</ref>
The conjecture is known to hold for certain special classes of convex bodies, including, in dimension three, centrally symmetric polyhedra and [[Surface of constant width|bodies of constant width]].<ref name="BMP"/> The number of copies needed to cover any [[zonotope]] (other than a parallelepiped) is at most <math>(3/4)2^n</math>, while for bodies with a smooth surface (that is, having a single tangent plane per boundary point), at most <math>n+1</math> smaller copies are needed to cover the body, as [[Friedrich Wilhelm Levi|Levi]] already proved.<ref name="BMP"/>
==See also==
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