A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2018; you can also visit the original URL.
The file type is `application/pdf`

.

## Filters

##
###
Pick's Theorem via Minkowski's Theorem

2007
*
The American mathematical monthly
*

In this note, we present a new proof of Pick's

doi:10.1080/00029890.2007.11920465
fatcat:ikey2zh7ufgntkrwrc22yffdie
*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

1988
*
Journal of Number Theory
*

In spite of its simple nature,

doi:10.1016/0022-314x(88)90090-x
fatcat:helxg7zuszgq5ji6udouj4xy3a
*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]

2020
*
arXiv
*
pre-print

In this paper we also derive applications of this

arXiv:2010.00245v1
fatcat:vhvqw3unkrdsbplkxjf3ug26ly
*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

1958
*
Proceedings of the American Mathematical Society
*

A

doi:10.1090/s0002-9939-1958-0093515-5
fatcat:6e7m6zzgbrbebprq35ihyt6koi
*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

1961
*
Proceedings of the American Mathematical Society
*

This proof can be readily extended to

doi:10.1090/s0002-9939-1961-0136578-0
fatcat:ikslozss5jfnjptvxekh76cn6m
*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

1958
*
Proceedings of the American Mathematical Society
*

A

doi:10.2307/2032986
fatcat:zgzgdbvu2zas5fsdrmv7qt6geq
*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]

2017
*
arXiv
*
pre-print

We will give a new proof to

arXiv:1608.03935v2
fatcat:lozrbxhxmndrjfnqomhu2j34z4
*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

2004
*
Indagationes mathematicae
*

The assumption that limt~0 ~b(t) = 0 in

doi:10.1016/s0019-3577(04)90006-7
fatcat:palmubr6snechm5pq4u2j2nipm
*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

doi:10.1016/s0012-365x(96)00119-7
fatcat:rbjcgda52feazi3dcvjbwiw4lq
*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

1959
*
Canadian mathematical bulletin
*

The above inequality leads at once to the estimate of f(n) in the

doi:10.4153/cmb-1959-019-3
fatcat:z5pvd3usgrgzxklbsgaum47u74
*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

1978
*
Bulletin of the Australian Mathematical Society
*

It is known that

doi:10.1017/s0004972700008261
fatcat:o5angkec5jhgpdg34gy2hgceom
*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

2008
*
European journal of combinatorics (Print)
*

Klee extended a well-known

doi:10.1016/j.ejc.2008.01.021
fatcat:i4hqepqzjfhp3pvnvlrgcx3bxi
*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

2016
*
Mathematika
*

One of the most fruitful results from

doi:10.1112/s002557931500042x
fatcat:qt3nwyxstzdvhmwb76x4vnswzq
*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]

2017
*
arXiv
*
pre-print

Finally, we describe two types of uniqueness results which follow from the extension of

arXiv:1607.06531v4
fatcat:zwagqckk5vc47hrpsnphvn72my
*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

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*. ...

« Previous

*Showing results 1 — 15 out of 10,924 results*