Covering Minima and Lattice-Point-Free Convex Bodies

1988 ◽  
Vol 128 (3) ◽  
pp. 577 ◽  
Author(s):  
Ravi Kannan ◽  
Laszlo Lovasz
Keyword(s):  
Lattice Point ◽  
Convex Bodies

Minimal width and diameter of lattice‐point‐free convex bodies

Mathematika ◽  
1981 ◽  
Vol 28 (2) ◽  
pp. 255-264 ◽  
Author(s):  
P. McMullen ◽  
J. M. Wills
Keyword(s):  
Lattice Point ◽  
Convex Bodies ◽  
Minimal Width

Note on lattice-point-free convex bodies

1998 ◽  
Vol 126 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Poh Wah Awyong ◽  
Martin Henk ◽  
Paul R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Bodies

scholarly journals Lattice Point Inequalities for Centered Convex Bodies

2016 ◽  
Vol 30 (2) ◽  
pp. 1148-1158
Author(s):  
Sören Lennart Berg ◽  
Martin Henk
Keyword(s):  
Lattice Point ◽  
Convex Bodies

scholarly journals Area-diameter and area-width relations for covering plane sets

1995 ◽  
Vol 51 (1) ◽  
pp. 163-169 ◽  
Author(s):  
Salvatore Vassallo
Keyword(s):  
Lattice Point ◽  
Convex Bodies ◽  
Plane Lattice

An area-diameter relation and an area-width relation for plane lattice-point-free-convex bodies is proved. This implies relations on covering sets with respect to general lattices.

scholarly journals On Counterexamples to a Conjecture of Wills and Ehrhart Polynomials whose Roots have Equal Real Parts

10.37236/3757 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Matthias Henze
Keyword(s):  
Lattice Point ◽  
Convex Bodies ◽  
Discrete Analog ◽  
Side Length ◽  
Ehrhart Polynomials ◽  
Lattice Polytopes ◽  
Lattice Polytope ◽  
Positive Side ◽  
Reflexive Polytopes

As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a $0$-symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side length two. We discuss several counterexamples to this conjecture and, on the positive side, we identify a family of lattice polytopes that fulfill the claimed inequalities. This family is related to the recently introduced class of $l$-reflexive polytopes.

A New Condition of Formation and Stablity of All Crystalline Systems in a Good Agreement with Experimental Data

2015 ◽  
Vol 11 (3) ◽  
pp. 3224-3228
Author(s):  
Tarek El-Ashram
Keyword(s):  
Experimental Data ◽  
Brillouin Zone ◽  
Electron Concentration ◽  
Free Electron ◽  
Lattice Point ◽  
Valence Electron ◽  
Fermi Gas ◽  
Valence Electron Concentration ◽  
Fermi Sphere ◽  
Good Agreement

In this paper we derived a new condition of formation and stability of all crystalline systems and we checked its validity andit is found to be in a good agreement with experimental data. This condition is derived directly from the quantum conditionson the free electron Fermi gas inside the crystal. The new condition relates both the volume of Fermi sphere VF andvolume of Brillouin zone VB by the valence electron concentration VEC as ;𝑽𝑭𝑽𝑩= 𝒏𝑽𝑬𝑪𝟐for all crystalline systems (wheren is the number of atoms per lattice point).

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