Hadwiger conjecture (combinatorial geometry): Difference between revisions

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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>{{Cite journal harvtxt|last=Huang |first=Han |last2=Slomka |first2=Boaz A. |last3=Tkocz |first3=Tomasz |last4=Vritsiou |first4=Beatrice-Helen |date=2021-08-11 |title=Improved bounds for Hadwiger’s covering problem via thin-shell estimates |url=https://ems.press/journals/jems/articles/2252801 |journal=Journal of the European Mathematical Society |language=en |volume=24 |issue=4 |pages=1431–1448 |doi=10.4171/jems/1132 |issn=1435-98552022}}</ref>

:<math>\displaystyle \binom{2n}{n}\exp(-c\sqrt{n})</math>.

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=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}}.

*{{Citation |last=Huang, |first=Han, |last2=Slomka, |first2=Boaz A., |last3=Tkocz, |first3=Tomasz, & |last4=Vritsiou, |first4=Beatrice-Helen (2021).|date=2022 |title=Improved bounds for Hadwiger’s covering problem via thin-shell estimates. ''|journal=[[Journal of the European Mathematical Society]]'' |language=en |volume=24 |issue=4 |pages=1431–1448 |doi=10.4171/jems/1132 |doi-access=free |issn=1435-9855}}

*{{citation|first=Marek|last=Lassak|year=1988|title=Covering the boundary of a convex set by tiles|journal=[[Proceedings of the American Mathematical Society]]|volume=104|issue=1|pages=269–272|mr=0958081|doi=10.1090/s0002-9939-1988-0958081-7|doi-access=free}}.

*{{citation|first=Friedrich Wilhelm|last=Levi|authorlink=Friedrich Wilhelm Levi|title=Überdeckung eines Eibereiches durch Parallelverschiebungen seines offenen Kerns|journal=[[Archiv der Mathematik]]|volume=6|year=1955|pages=369–370|issue=5|doi=10.1007/BF01900507|doi-access=free|s2cid=121459171}}.