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==Known results==
The two-dimensional case was settled by {{harvtxt|Levi|1955}}: every two-dimensional bounded convex set may be covered with four smaller copies of itself, with the fourth copy needed only in the case of parallelograms. However, the conjecture remains open in higher dimensions except for some special cases. The best known asymptotic upper bound on the number of smaller copies needed to cover a given body is
:<math>\displaystyle 4^n (5n\ln n).</math> {{Citation needed}}
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>
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