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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.  ...

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.  ...

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).  ...

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.  ...

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.  ...

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.  ...

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 .  ...

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.  ...

Two consequences of Minkowski's 2n theorem

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.  ...

Minkowski's theorem with curvature limitations. I

Z. A. Melzak
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).  ...

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.  ...

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.  ...

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.  ...

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.  ...

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.  ...
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