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→Known results: Corrected an otherwise wrong statement. Tags: Mobile edit Mobile web edit |
The best known bound has been improved. I've updated the article to reflect this. |
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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<ref>{{harvtxt|
:<math>\displaystyle
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>
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*{{citation|first1=Peter|last1=Brass|first2=William|last2=Moser|
first3=János|last3=Pach|author3-link=János Pach|contribution=3.3 Levi–Hadwiger Covering Problem and Illumination|pages=136–142|title=Research Problems in Discrete Geometry|publisher=Springer-Verlag|year=2005}}.
*{{citation|first1=Marcelo|last1=Campos|first2=Peter|last2=van Hintum|first3=Robert|last3=Morris|author3-link=Robert Morris (mathematician)|first4=Marius|last4=Tiba|title=Towards Hadwiger’s Conjecture via Bourgain Slicing|journal=[[International Mathematics Research Notices]]|doi=10.1093/imrn/rnad198|year=2023}}.
*{{citation|first1=Israel Ts.|last1=Gohberg|authorlink1=Israel Gohberg|first2=Alexander S.|last2=Markus|language= Russian|year=1960|title=A certain problem about the covering of convex sets with homothetic ones|journal=Izvestiya Moldavskogo Filiala Akademii Nauk SSSR|volume=10|issue=76|pages=87–90}}.
*{{citation|first=Hugo|last=Hadwiger|authorlink=Hugo Hadwiger|year=1957|title=Ungelöste Probleme Nr. 20|journal=Elemente der Mathematik|volume=12|pages=121}}.
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