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# Minkowski's Conjecture

Minkowski's conjecture states that every lattice tiling of by unit hypercubes contains two hypercubes that meet in an -dimensional face. Minkowski first considered the problem in 1896, when he stated it as a theorem whose proof would be provided later. However, it subsequently appeared as on open problem in Minkowski's 1907 book, suggesting the purported proof was erroneous. The conjecture was subsequently proved in eight and fewer dimensions by Peron (1940) and in general by Hajós (1942).

Keller's conjecture is a generalization of Minkowski's conjecture (which however is known to be true only in dimensions six and less and to be false in at least dimensions 8, 10, and 12).

Keller's Conjecture

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## References

Debroni, J.; Eblen, J. D.; Langston, M. A.; Shor, P.; Myrvold, W.; and Weerapurage, D. "A Complete Resolution of the Keller Maximum Clique Problem." Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms. pp. 129-135, 2011. http://www.siam.org/proceedings/soda/2011/SODA11_011_debronij.pdf.Hajós, G. "Über einfache und mehrfache Bedeckung des -dimensionalen Raumes mit einen Würfelgitter." Math. Z. 47, 427-467, 1942.Mackey, J. "A Cube Tiling of Dimension Eight with No Facesharing." Disc. Comput. Geom. 28, 275-279, 2002.Minkowski, H. Diophantische Approximationen. Leipzig, Germany: Teubner, 1907.Peron, O. "Über lückenlose Ausfüllung des -dimensionalen raumes durch kongruente Würfel I & II." Math. Z. 46, 1-26 and 161-180, 1940.

## Cite this as:

Weisstein, Eric W. "Minkowski's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MinkowskisConjecture.html