Lattice packing and covering of convex bodies

2011 ◽  
Vol 275 (1) ◽  
pp. 229-238 ◽  
Author(s):  
Peter M. Gruber
Keyword(s):  
Convex Bodies ◽  
Lattice Packing ◽  
Packing And Covering

Extremum properties of lattice packing and covering with circles

2016 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter M. Gruber
Keyword(s):  
Packing Density ◽  
Quadratic Forms ◽  
Circle Packing ◽  
Convex Bodies ◽  
Positive Definite ◽  
Smooth Convex ◽  
Packing And Covering ◽  
Classical Criterion ◽  
Special Case ◽  
Lattice Packings

AbstractRecent results on extremum properties of the density of lattice packings of smooth convex bodies and balls extend and refine Voronoĭ’s classical criterion for balls. This article treats in more detail the special case of lattice packings and coverings with circular discs. The aim is to determine those lattices for which the densities of the corresponding packings and coverings with circular discs, and certain products and quotients thereof, are semi-stationary, stationary, extreme, and ultra-extreme. The latter notion is a sharper version of extremality. It turns out that in all cases where solutions exist, the regular hexagonal lattices are solutions. Unexpectedly, in a few cases the square lattices and in one case special parallelogram lattices are solutions too. A further surprise is the fact that the lattices forwhich the circle packing density is extreme coincide with the lattices with ultra-extreme density. For semi-stationarity, stationarity and ultra-extremality the duality between packing and covering results breaks down. All results may be interpreted in terms of binary positive definite quadratic forms.

Excess entropy of the systems of molecules differing in size and shape

1983 ◽  
Vol 48 (1) ◽  
pp. 192-198 ◽  
Author(s):  
Tomáš Boublík
Keyword(s):  
Equation Of State ◽  
Hard Spheres ◽  
Excess Entropy ◽  
Convex Bodies ◽  
Pure Substance ◽  
Liquid Equilibrium ◽  
Simple Rules ◽  
Different Shapes ◽  
The Mean

The excess entropy of mixing of mixtures of hard spheres and spherocylinders is determined from an equation of state of hard convex bodies. The obtained dependence of excess entropy on composition was used to find the accuracy of determining ΔSE from relations employed for the correlation and prediction of vapour-liquid equilibrium. Simple rules were proposed for establishing the mean parameter of nonsphericity for mixtures of hard bodies of different shapes allowing to describe the P-V-T behaviour of solutions in terms of the equation of state fo pure substance. The determination of ΔSE by means of these rules is discussed.

scholarly journals Packing and Covering Balls in Graphs Excluding a Minor

COMBINATORICA ◽  
2021 ◽  
Author(s):  
Nicolas Bousquet ◽  
Wouter Cames Van Batenburg ◽  
Louis Esperet ◽  
Gwenaël Joret ◽  
William Lochet ◽  
...  
Keyword(s):  
Packing And Covering ◽  
A Minor

scholarly journals Some Hermite–Hadamard type integral inequalities for convex functions defined on convex bodies in ℝn\mathbb{R}^{n}

2020 ◽  
Vol 26 (1) ◽  
pp. 67-77 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Euclidean Space ◽  
Convex Functions ◽  
Convex Bodies ◽  
Integral Inequalities ◽  
Divergence Theorem ◽  
Several Variables ◽  
Type Integral ◽  
Bounded Convex

AbstractIn this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space {\mathbb{R}^{n}} for any {n\geq 2}.

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