scholarly journals Successive minima, intrinsic volumes, and lattice determinants

1995 ◽  
Vol 13 (2) ◽  
pp. 233-239 ◽  
Author(s):  
U. Schnell
Mathematika ◽  
1993 ◽  
Vol 40 (1) ◽  
pp. 144-147 ◽  
Author(s):  
U. Schnell ◽  
J. M. Wills

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

2009 ◽  
Vol 410 (18) ◽  
pp. 1648-1665 ◽  
Author(s):  
Johannes Blömer ◽  
Stefanie Naewe

1969 ◽  
Vol 10 (1-2) ◽  
pp. 177-181 ◽  
Author(s):  
I. Danicic

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


Author(s):  
Mikołaj Fraczyk ◽  
Gergely Harcos ◽  
Péter Maga

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.


2011 ◽  
Vol 28 (2) ◽  
pp. 77 ◽  
Author(s):  
Joachim Ohser ◽  
Werner Nagel ◽  
Katja Schladitz

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.


1995 ◽  
Vol 38 (2) ◽  
pp. 156-166 ◽  
Author(s):  
Károly Bőrőczky ◽  
Martin Henk

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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