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Pick's Theorem via Minkowski's Theorem

M. Ram Murty, Nithum Thain
2007 The American mathematical monthly  
In this note, we present a new proof of Pick's theorem via Minkowski's convex body theorem.  ...  We now use Minkowski's theorem to prove Pick's theorem for the case of elementary triangles.  ... 
doi:10.1080/00029890.2007.11920465 fatcat:ikey2zh7ufgntkrwrc22yffdie

Modifying Minkowski's theorem

Paul R Scott
1988 Journal of Number Theory  
In spite of its simple nature, Minkowski's theorem is a powerful and important result.  ...  Now Minkowski's theorem states that if K is a convex body which is symmetric about the origin 0, and if K contains no nonzero points of the lattice A, then the volume V(K) of K satisfies V(K) < 2" d(n)  ...  The number 2 in the denominator appears to be significant for Minkowski's theorem.  ... 
doi:10.1016/0022-314x(88)90090-x fatcat:helxg7zuszgq5ji6udouj4xy3a

Minkowski's Lattice Point Theorem [article]

Sourangshu Ghosh
2020 arXiv   pre-print
In this paper we also derive applications of this theorem in proving important theorems in number theory.  ...  This paper formulates the elegant theorem on the existence of a nonzero lattice point in any convex symmetric body of sufficiently large volume proved by Minkowski.  ...  The following theorem was derived by Dirichlet using rather elementary methods. However, it can also be viewed as a direct corollary of Minkowski's theorem. Theorem 4(Dirichlet).  ... 
arXiv:2010.00245v1 fatcat:vhvqw3unkrdsbplkxjf3ug26ly

On a theorem of Minkowski's

A. C. Woods
1958 Proceedings of the American Mathematical Society  
A theorem of Minkowski's (1) is that there are at most 2n -1 pairs of points ±X of A that lie on the boundary of K.  ...  I note here that this is a special case of a theorem which applies to any lattice. Let K be as above and let A be an arbitrary lattice in Rn, so not necessarily i^-admissible.  ... 
doi:10.1090/s0002-9939-1958-0093515-5 fatcat:6e7m6zzgbrbebprq35ihyt6koi

Minkowski's theorem on nonhomogeneous approximation

Ivan Niven
1961 Proceedings of the American Mathematical Society  
This proof can be readily extended to Minkowski's theorem on the product of two linear forms, as will be shown elsewhere.  ...  But at most one pair can give F= 1/4, because Oxi + yi + y = ± (4xi)_1 and 0x2 + y2 + y = ± (4x2)-1 i96i] MINKOWSKI'S THEOREM ON APPROXIMATION 993 and so (A2) must hold for u = n or u = n + l.  ... 
doi:10.1090/s0002-9939-1961-0136578-0 fatcat:ikslozss5jfnjptvxekh76cn6m

On a Theorem of Minkowski's

A. C. Woods
1958 Proceedings of the American Mathematical Society  
A theorem of Minkowski's (1) is that there are at most 2n -1 pairs of points ±X of A that lie on the boundary of K.  ...  I note here that this is a special case of a theorem which applies to any lattice. Let K be as above and let A be an arbitrary lattice in Rn, so not necessarily i^-admissible.  ... 
doi:10.2307/2032986 fatcat:zgzgdbvu2zas5fsdrmv7qt6geq

Minkowski's theorem on independent conjugate units [article]

Shabnam Akhtari, Jeffrey D. Vaaler
2017 arXiv   pre-print
We will give a new proof to Minkowski's theorem and show that there exists a Minkowski unit $\beta \in l$ such that the Weil height of $\beta$ is comparable with the sum of the heights of a fundamental  ...  Here we give a proof of Minkowski's theorem which includes a bound on the index [F l : B], and a bound on the absolute logarithmic Weil height h(β) of the Minkowski unit β. Theorem 1.1.  ...  In general the Galois group Aut(l/Q) does not act on the subgroup E l/k , and therefore the simplest analogue of Minkowski's theorem cannot hold in E l/k .  ... 
arXiv:1608.03935v2 fatcat:lozrbxhxmndrjfnqomhu2j34z4

The converse theorem for Minkowski's inequality

Janusz Matkowski
2004 Indagationes mathematicae  
The assumption that limt~0 ~b(t) = 0 in Theorem 1 is superfluous.  ...  As an immediate consequence of Proposition 1 we obtain the following Theorem 1.  ... 
doi:10.1016/s0019-3577(04)90006-7 fatcat:palmubr6snechm5pq4u2j2nipm

Two consequences of Minkowski's 2n theorem

Xueqing Tang, Adi Ben-Israel
1997 Discrete Mathematics  
Results Theorem 1. Let A = (aij) E ~rxn have rank r < n, and let b = (bl) E ~m be [2, p. 17]). Let the matrix (alj) ~ ~n×n be nonsingular, and let b = (bi) ~ •m be a positive vector.  ...  We deal here with two of its well-known consequences: The linear form theorem of Minkowski (Minkowski [4] and Erd6s et al. Here k, n denotes the index set {k, k + 1, ..., n}, for integers k ~< n.  ... 
doi:10.1016/s0012-365x(96)00119-7 fatcat:rbjcgda52feazi3dcvjbwiw4lq

Minkowski's theorem with curvature limitations. I

Z. A. Melzak
1959 Canadian mathematical bulletin  
The above inequality leads at once to the estimate of f(n) in the theorem.  ...  THEOREM 1. Let K be a maximal r-region in E n . Then K is the intersection of f(n) solid n-spheres of radius r, where f(n) is an even integer and f(n) < 2 n n(n-2)l +n -2/(n-l).  ... 
doi:10.4153/cmb-1959-019-3 fatcat:z5pvd3usgrgzxklbsgaum47u74

A new extension of Minkowski's Theorem

P.R. Scott
1978 Bulletin of the Australian Mathematical Society  
It is known that Minkowski's Theorem in the plane remains true for a large class of non-symmetric sets (for example, [/]), but the following simple result appears to have been overlooked.  ...  This completes the proof of the theorem.  ... 
doi:10.1017/s0004972700008261 fatcat:o5angkec5jhgpdg34gy2hgceom

Minkowski's theorem for arbitrary convex sets

Tudor Zamfirescu
2008 European journal of combinatorics (Print)  
Klee extended a well-known theorem of Minkowski to non-compact convex sets. We generalize Minkowski's theorem to convex sets which are not necessarily closed.  ...  The following example illustrates the larger applicability of Theorem 1, compared with Minkowski's theorem. Example.  ...  Klee's generalization [2] of Minkowski's theorem can be strengthened in the same way. Theorem 2. Let A be convex and line-free.  ... 
doi:10.1016/j.ejc.2008.01.021 fatcat:i4hqepqzjfhp3pvnvlrgcx3bxi

A GENERALIZATION OF THE DISCRETE VERSION OF MINKOWSKI'S FUNDAMENTAL THEOREM

Bernardo González Merino, Matthias Henze
2016 Mathematika  
One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so-called first fundamental theorem.  ...  In particular, we are grateful for pointing us to the connection with Helly numbers of families of S-convex sets and to the proof of Theorem 1.2.  ...  One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so-called first fundamental theorem.  ... 
doi:10.1112/s002557931500042x fatcat:qt3nwyxstzdvhmwb76x4vnswzq

An extension of Minkowski's theorem and its applications to questions about projections for measures [article]

Galyna V. Livshyts
2017 arXiv   pre-print
Finally, we describe two types of uniqueness results which follow from the extension of Minkowski's theorem.  ...  In this manuscript we prove an extension of Minkowski's theorem. Consider a measure $\mu$ on $\mathbb{R}^n$ with positive degree of concavity and positive degree of homogeneity.  ...  EXTENSION OF THE MINKOWSKI'S EXISTENCE THEOREM. This section is dedicated to proving an extension of Minkowski's existence theorem.  ... 
arXiv:1607.06531v4 fatcat:zwagqckk5vc47hrpsnphvn72my

Minkowski's Convex Body Theorem and Integer Programming

Ravi Kannan
1987 Mathematics of Operations Research  
Minkowski's theorem implies that this set of candidates suffices.  ...  This paper uses several concepts and results from Geometry of Numbers, the most crucial of them being Minkowski's convex body theorem.  ...  Of course, v -u is in Z n proving the theorem. • I will generally only use the following direct consequence of Minkowski's theorem.  ... 
doi:10.1287/moor.12.3.415 fatcat:o2cmwczgrjbyfe7aic5fqteklq
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