Proceedings - Mathematical Sciences

, Volume 126, Issue 4, pp 501–548 | Cite as

On conjectures of Minkowski and Woods for n = 9

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Abstract

Let ℝ n be the n-dimensional Euclidean space with O as the origin. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of ∧ other than O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in ℝ n of radius \( \sqrt {n/4}\) contains a point of ∧. This is known to be true for n≤8. Here we prove a more general conjecture of Woods for n = 9 from which this conjecture follows in ℝ9. Together with a result of McMullen (J. Amer. Math. Soc. 18 (2005) 711–734), the long standing conjecture of Minkowski follows for n = 9.

Keywords

Lattice covering non-homogeneous product of linear forms critical determinant. 

Mathematics Subject Classification.

11H31 11H46 11J20 11J37 52C15. 
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Notes

Acknowledgements

The authors are grateful to Professors R. P. Bambah, R. J. Hans-Gill and Ranjeet Sehmi for various discussions throughout the preparation of this paper. The authors would like to thank the anonymous referee for a very meticulous reading of the manuscript and checking the massive calculations performed in it. The first author acknowledges the support of CSIR, India. The paper forms a part of her Ph.D. dissertation accepted by Panjab University, Chandigarh.

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Copyright information

© Indian Academy of Sciences 2016

Authors and Affiliations

  1. 1.Centre for Advanced Study in MathematicsPanjab UniversityChandigarhIndia

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