scholarly journals Improving Rogers’ Upper Bound for the Density of Unit Ball Packings via Estimating the Surface Area of Voronoi Cells from Below in Euclidean \sl d -Space for All \sl d ≥ \bf 8

2002 ◽  
Vol 28 (1) ◽  
pp. 75-106 ◽  
Author(s):  
Bezdek
Keyword(s):  
Unit Ball ◽  
Surface Area ◽  
Upper Bound ◽  
Voronoi Cells ◽  
Ball Packings

scholarly journals Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

2008 ◽  
Vol 60 (1) ◽  
pp. 3-32 ◽  
Author(s):  
Károly Böröczky ◽  
Károly J. Böröczky ◽  
Carsten Schütt ◽  
Gergely Wintsche
Keyword(s):  
Unit Ball ◽  
Surface Area ◽  
Convex Bodies ◽  
Extreme Points ◽  
Thin Shells ◽  
Minimal Volume ◽  
Asymptotic Formulae ◽  
Mean Width

AbstractGiven r > 1, we consider convex bodies in En which contain a fixed unit ball, and whose extreme points are of distance at least r from the centre of the unit ball, and we investigate how well these convex bodies approximate the unit ball in terms of volume, surface area and mean width. As r tends to one, we prove asymptotic formulae for the error of the approximation, and provide good estimates on the involved constants depending on the dimension.

scholarly journals Top dense hyperbolic ball packings and coverings for complete coxeter orthoscheme groups

2018 ◽  
Vol 103 (117) ◽  
pp. 129-146
Author(s):  
Emil Molnár ◽  
Jenő Szirmai
Keyword(s):  
Hyperbolic Space ◽  
Coxeter Group ◽  
Upper Bound ◽  
Dihedral Angles ◽  
Ball Packings ◽  
Ball Packing ◽  
Packing Densities

In n-dimensional hyperbolic space Hn (n > 2), there are three types of spheres (balls): the sphere, horosphere and hypersphere. If n = 2, 3 we know a universal upper bound of the ball packing densities, where each ball?s volume is related to the volume of the corresponding Dirichlet-Voronoi (D-V) cell. E.g., in H3 a densest (not unique) horoball packing is derived from the {3,3,6} Coxeter tiling consisting of ideal regular simplices T? reg with dihedral angles ?/3. The density of this packing is ??3 ? 0.85328 and this provides a very rough upper bound for the ball packing densities as well. However, there are no ?essential" results regarding the ?classical" ball packings with congruent balls, and for ball coverings either. The goal of this paper is to find the extremal ball arrangements in H3 with ?classical balls". We consider only periodic congruent ball arrangements (for simplicity) related to the generalized, so-called complete Coxeter orthoschemes and their extended groups. In Theorems 1.1 and 1.2 we formulate also conjectures for the densest ball packing with density 0.77147... and the loosest ball covering with density 1.36893..., respectively. Both are related with the extended Coxeter group (5,3,5) and the so-called hyperbolic football manifold. These facts can have important relations with fullerenes in crystallography.

On Benson’s Definition of Area in Minkowski Space

1999 ◽  
Vol 42 (2) ◽  
pp. 237-247 ◽  
Author(s):  
A. C. Thompson
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Minkowski Space ◽  
Surface Area ◽  
Isoperimetric Problem ◽  
Dimensional Banach Space ◽  
Finite Dimensional ◽  
The One ◽  
Definition Of ◽  
Finite Dimensional Banach Space

AbstractLet (X, ‖ . ‖) be a Minkowski space (finite dimensional Banach space) with unit ball B. Various definitions of surface area are possible in X. Here we explore the one given by Benson [1], [2]. In particular, we show that this definition is convex and give details about the nature of the solution to the isoperimetric problem.

scholarly journals Large Steklov eigenvalues via homogenisation on manifolds

2021 ◽  
Author(s):  
Alexandre Girouard ◽  
Jean Lagacé
Keyword(s):  
Free Boundary ◽  
Unit Ball ◽  
Lower Bounds ◽  
Upper Bound ◽  
Minimal Surfaces ◽  
Constant Weight ◽  
Steklov Eigenvalue ◽  
Laplace Eigenvalues ◽  
Steklov Eigenvalues ◽  
Steklov Eigenfunctions

AbstractUsing methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev’s upper bound of $$8\pi $$ 8 π for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than $$2\pi $$ 2 π . This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus zero in the unit ball with even larger area. We prove that the first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they are consistent with a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois–El Soufi–Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.

scholarly journals Large Poisson-Voronoi cells and Crofton cells

2004 ◽  
Vol 36 (03) ◽  
pp. 667-690 ◽  
Author(s):  
Daniel Hug ◽  
Matthias Reitzner ◽  
Rolf Schneider
Keyword(s):  
Euclidean Space ◽  
Surface Area ◽  
High Probability ◽  
Voronoi Tessellation ◽  
Voronoi Cells ◽  
Typical Cell ◽  
Zero Cell

It is proved that the shape of the typical cell of a stationary Poisson-Voronoi tessellation in Euclidean space, under the condition that the volume of the typical cell is large, must be close to spherical, with high probability. The same holds if the volume is replaced by the surface area or other suitable functionals. Similar results are established for the zero cell of a stationary and isotropic Poisson hyperplane tessellation.

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