Successive-minima-type inequalities

Betke, U.; Henk, Martin; Wills, J. M.

doi:10.1007/bf02189316

Cited by 54 publications

(55 citation statements)

References 2 publications

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“…It should be noted that the conjecture above, if true, would imply Minkowski's second theorem on successive minima, using a simple argument involving the definition of the Riemann integral [1]. Betke, Henk, and Wills proved that Conjecture 1.1 holds roughly up to a factor of d!.…”

Section: Introductionmentioning

confidence: 95%

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An Optimization Problem Related to Minkowski’s Successive Minima

Malikiosis

2009

Discrete Comput Geom

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The purpose of this paper is to establish an inequality connecting the lattice point enumerator of a 0-symmetric convex body with its successive minima. To this end, we introduce an optimization problem whose solution refines former methods, thus producing a better upper bound. In particular, we show that an analogue of Minkowski's second theorem on successive minima with the volume replaced by lattice point enumerator is true up to an exponential factor, whose base is approximately 1.64.

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Section: Introductionmentioning

confidence: 95%

“…In 1993, Betke, Henk, and Wills [1] stated analogues of Minkowski's theorems for the lattice point enumerator, instead of the volume. Their first theorem is the following:…”

Section: Introductionmentioning

confidence: 99%

An Optimization Problem Related to Minkowski’s Successive Minima

Malikiosis

2009

Discrete Comput Geom

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“…(5) Here, the floor function x and the ceiling function x of a real number x denote, as usual, the largest integer smaller than or equal to x, and the smallest integer bigger than or equal to x, respectively. Clearly, (2) and (3) are special cases of (4) and (5), respectively.…”

Section: Introductionmentioning

confidence: 99%

“…The first is an application of an elegant congruence argument that lies behind many pertinent results in the geometry of numbers (cf. [5,14,20] or [9, Sect. 30/31]).…”

Section: Introductionmentioning

confidence: 99%

A Generalization of the Discrete Version of Minkowski's Fundamental Theorem

Merino

Henze

2016

Mathematika

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In memory of Hermann Minkowski on the occasion of his 150th birthday Abstract. One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of a 0-symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of a 0-symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.

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“…The equality case in (1.1) has been characterized by Draisma, Nill and McAllister [7]. Furthermore, it was pointed out by Betke et al [5] that Minkowski's bounds can easily be extended to arbitrary o-symmetric convex bodies via the first successive minimum λ 1 (K) of K, which, for latter purposes, we define here for any convex body with 0 ∈ int (K) :…”

Section: Introductionmentioning

confidence: 99%

Lattice Point Inequalities for Centered Convex Bodies

Berg¹,

Henk²

2016

SIAM J. Discrete Math.

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Abstract. We study upper bounds on the number of lattice points for convex bodies having their centroid at the origin. For the family of simplices as well as in the planar case we obtain best possible results. For arbitrary convex bodies we provide an upper bound, which extends the centrally symmetric case and which, in particular, shows that the centroid assumption is indeed much more restrictive than an assumption on the number of interior lattice points even for the class of lattice polytopes.

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Successive-minima-type inequalities

Abstract: We show analogues of Minkowski's theorem on successive minima, where the volume is replaced by the lattice point enumerator. We further give analogous results to some recent theorems by Kannan and Lov/tsz on covering minima.

Cited by 54 publications

References 2 publications

An Optimization Problem Related to Minkowski’s Successive Minima

An Optimization Problem Related to Minkowski’s Successive Minima

A Generalization of the Discrete Version of Minkowski's Fundamental Theorem

Lattice Point Inequalities for Centered Convex Bodies

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