scholarly journals Successive minima, intrinsic volumes, and lattice determinants

1995 ◽  
Vol 13 (2) ◽  
pp. 233-239 ◽  
Author(s):  
U. Schnell
Keyword(s):  
Intrinsic Volumes ◽  
Successive Minima

On successive minima and intrinsic volumes

Mathematika ◽  
1993 ◽  
Vol 40 (1) ◽  
pp. 144-147 ◽  
Author(s):  
U. Schnell ◽  
J. M. Wills
Keyword(s):  
Intrinsic Volumes ◽  
Successive Minima

scholarly journals Random ball-polyhedra and inequalities for intrinsic volumes

2016 ◽  
Vol 182 (3) ◽  
pp. 709-729 ◽  
Author(s):  
Grigoris Paouris ◽  
Peter Pivovarov
Keyword(s):  
Intrinsic Volumes

scholarly journals Sampling methods for shortest vectors, closest vectors and successive minima

2009 ◽  
Vol 410 (18) ◽  
pp. 1648-1665 ◽  
Author(s):  
Johannes Blömer ◽  
Stefanie Naewe
Keyword(s):  
Sampling Methods ◽  
Successive Minima

scholarly journals An elementary proof of Minkowski's second inequality

1969 ◽  
Vol 10 (1-2) ◽  
pp. 177-181 ◽  
Author(s):  
I. Danicic
Keyword(s):  
Euclidean Space ◽  
Elementary Proof ◽  
Content Volume ◽  
Lattice Points ◽  
The Body ◽  
Dimensional Euclidean Space ◽  
Open Convex ◽  
Successive Minima ◽  
Linearly Independent ◽  
Independent Lattice

Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously

scholarly journals Counting Bounded Elements of a Number Field

2020 ◽  
Author(s):  
Mikołaj Fraczyk ◽  
Gergely Harcos ◽  
Péter Maga
Keyword(s):  
Group Theory ◽  
Number Field ◽  
Geometry Of Numbers ◽  
Combine Group ◽  
Successive Minima ◽  
Ideal Lattices ◽  
Linearly Independent ◽  
Ramification Theory

Abstract We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices. Our arguments combine group theory, ramification theory, and the geometry of numbers.

scholarly journals MILES FORMULAE FOR BOOLEAN MODELS OBSERVED ON LATTICES

2011 ◽  
Vol 28 (2) ◽  
pp. 77 ◽  
Author(s):  
Joachim Ohser ◽  
Werner Nagel ◽  
Katja Schladitz
Keyword(s):  
Mean Curvature ◽  
Surface Density ◽  
Euler Number ◽  
Volume Density ◽  
Random Sets ◽  
Boolean Models ◽  
Intrinsic Volumes ◽  
Binary Images ◽  
Density Surface ◽  
The Mean

The densities of the intrinsic volumes – in 3D the volume density, surface density, the density of the integral of the mean curvature and the density of the Euler number – are a very useful collection of geometric characteristics of random sets. Combining integral and digital geometry we develop a method for efficient and simultaneous calculation of the intrinsic volumes of random sets observed in binary images in arbitrary dimensions. We consider isotropic and reflection invariant Boolean models sampled on homogeneous lattices and compute the expectations of the estimators of the intrinsic volumes. It turns out that the estimator for the surface density is proved to be asymptotically unbiased and thusmultigrid convergent for Boolean models with convex grains. The asymptotic bias of the estimators for the densities of the integral of the mean curvature and of the Euler number is assessed for Boolean models of balls of random diameters. Miles formulae with corresponding correction terms are derived for the 3D case.

Radii and the Sausage Conjecture

1995 ◽  
Vol 38 (2) ◽  
pp. 156-166 ◽  
Author(s):  
Károly Bőrőczky ◽  
Martin Henk
Keyword(s):  
Convex Hull ◽  
Intrinsic Volumes ◽  
Low Dimension ◽  
Extremal Properties

AbstractIn 1975, L. Fejes Toth conjectured that in Ed, d ≥ 5, the sausage arrangement is denser than any other packing of n unit balls. This has been known if the convex hull Cn of the centers has low dimension. In this paper, we settle the case when the inner m-radius of Cn is at least O(ln d/m). In addition, we consider the extremal properties of finite ballpackings with respect to various intrinsic volumes.

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