Talk:Minkowski–Hlawka theorem

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I guess the formulation of the Minkowski-Hlawka theorem should be strongly generalized from balls to central symmetric convex bodies. Cf. section 3.1 in "Sphere packings", Chuanming Zong,John Talbot. See http://www.amazon.com/Sphere-Packings-Universitext-Chuanming-Zong/dp/0387987940/ resp. google books on page 47. Shuber2 (talk) 10:54, 13 December 2009 (UTC)


In the introduction of Siegel, 1944, "A mean value theorem in geometry of numbers" one can read: "As a consequence Hlawka deduced an assertion of Minkowski which had remained unproven for more than fifty years: 'If B is an n-dimensional star domain of volume < zeta(n), then there exists a lattice of determinant 1 such that B does not contain any lattice point != 0.'

Hence, the formulation should even be extended to star-shaped sets, which includes balls as an extremely special case. However, I have also another concern on the current formulation on the theorem. In Gruber, "Convex and Discrete Geometry" the formulation is as follows: 'Let J be a Jordan measurable set in E^d with V(J) < 1. Then there is a lattice L in Ed with d(L) = 1 which contains no point of J, with the possible exception of o.'

Summarizing, there seem to be different formulation of this fundamental theorem. However, the current formulation is far too less general. Shuber2 (talk) 21:00, 17 December 2009 (UTC)