1 Introduction
A lattice is a discrete additive group of an euclidean space. Lattices are of great interest for discrete optimization, for example , the lattice of points of integral coordinates, is extensively used in integer programming. We analyze the relation between the convergence of a sequence of lattices and the set theoretic convergence of the corresponding Voronoi cells.
We prove that if is a sequence of full rank lattices from and  a full rank lattice also, then
where, for a given sequence of sets from
are the set theoretic limit inferior and limit superior.
2 Notations and definitions
Definition 1.
A lattice is a discrete additive group of , i. e., , and there exists an , such that . Its rank, , is the dimension of its spanned subspace, .
For every lattice there exist
independent vectors
in , such thatwhere . This a basis of . For a given base, if is the matrix whose columns are the basis vectors, the determinant of is
When the lattice has full rank, the determinant is the volume of the fundamental parallelepiped:
The fundamental parallelipiped has the following property: every vector in can be written uniquely as , where and . That is the spanned subspace can be tiled with copies of the fundamental parallelepiped centered in the vectors of the lattice. The fundamental parallelepiped depends on a particular basis but there is a similar polytope which can be uniquely associated with a lattice:
Definition 2.
Let be a full rank lattice. Its Voronoi cell is .
The Voronoi cell has a similar property of tessellation:
where the intersection of different tiles (cells) occurs only on their boundaries (frontiers) and .
The following two lattice parameters (see [2], [3]) illustrates the importance of the Voronoi cell. If is a lattice, the packing radius of , , is half the length of the shortest nonzero vector of , i. e.
The covering radius of is the smallest such that spheres of radius centered around all the points in cover the entire :
Definition 3.
Let and be full rank lattices in , we say that converges to , if, for each , there exists a basis of such that there exist the limits
and is a basis of .
3 Main result
Lemma 1.
Let be a full rank lattice. There exists such that and is a bounded set. (Consequence: is a polytope.)
proof: We prove first that is bounded; let be a basis of and . If , then , for some , and
Define ; obviously, . Let and , with . We have .
Lemma 2.
Let be a sequence of full rank lattices from . If , a full rank lattice from , then there exists such that and , .
proof: Define and ; we have . There exists such that , . Hence by choosing and , we have the desired property.
Theorem 1.
([2], Theorem 1, V.3) A necessary and sufficient condition that is that the following two conditions be both satisfied

if , there are points , for such that

if , there is a number and an integer , both depending on , such that
Where and are full rank lattices from .
Theorem 2.
Let be a sequence of full rank lattices from . If , a full rank lattice from , then .
proof: Suppose that is a basis of , , and is a basis of such that , . For every , there exists such that , and .
Let , i. e., there exists such that . Choose and ; we have and
hence .
We proved that . Now we prove that . Suppose not and let . It means that there exists an such that
Since , we have , with . By perturbing and decreasing (if necessary), we can suppose that
Using this property we can define an increasing sequence such that , for any ; hence there exist the points , such that , . Since , we can extract a convergent subsequence from , which, for the sake of simplicity, will be named in the same way: . Obviously , , and, by (ii) from Theorem 1,  a contradiction with the above property.
4 Conclusions
The converse of our main result it remains an open problem for now, but it can be proved (which is left for another technical report) that, if the extremal points of are contained in (when this happens, , starting from a certain ), then . It remains to see what happens when some of the extremal points of belong on the boundary of .
References
 [1] A. Barvinok. Combinatorics, geometry and complexity of integer points. electronic lecture notes. http://www.math.lsa.umich.edu/~barvinok/latticenotes669.pdf.
 [2] J. W. S. Cassels An introduction to the geometry of numbers. Springer Verlag, New York, 1997.
 [3] M. D. Sikiric, A. Tarlecki. Complexity and algorithms for computing Voronoi cells of lattices, Mathematics of Computation 78(267), 1713–1731, 2009. doi:10.1090/S0025571809022248.
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