1,734 research outputs found
On the convergence of the affine hull of the Chv\'atal-Gomory closures
Given an integral polyhedron P and a rational polyhedron Q living in the same
n-dimensional space and containing the same integer points as P, we investigate
how many iterations of the Chv\'atal-Gomory closure operator have to be
performed on Q to obtain a polyhedron contained in the affine hull of P. We
show that if P contains an integer point in its relative interior, then such a
number of iterations can be bounded by a function depending only on n. On the
other hand, we prove that if P is not full-dimensional and does not contain any
integer point in its relative interior, then no finite bound on the number of
iterations exists.Comment: 13 pages, 2 figures - the introduction has been extended and an extra
chapter has been adde
Successive Minima and Best Simultaneous Diophantine Approximations
We study the problem of best approximations of a vector by rational vectors of a lattice whose
common denominator is bounded. To this end we introduce successive minima for a
periodic lattice structure and extend some classical results from geometry of
numbers to this structure. This leads to bounds for the best approximation
problem which generalize and improve former results.Comment: 8 page
The isodiametric problem with lattice-point constraints
In this paper, the isodiametric problem for centrally symmetric convex bodies
in the Euclidean d-space R^d containing no interior non-zero point of a lattice
L is studied. It is shown that the intersection of a suitable ball with the
Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among
all bodies with the same volume. It is conjectured that these sets are the only
extremal bodies, which is proved for all three dimensional and several
prominent lattices.Comment: 12 pages, 4 figures, (v2) referee comments and suggestions
incorporated, accepted in Monatshefte fuer Mathemati
Lattice-point enumerators of ellipsoids
Minkowski's second theorem on successive minima asserts that the volume of a
0-symmetric convex body K over the covolume of a lattice \Lambda can be bounded
above by a quantity involving all the successive minima of K with respect to
\Lambda. We will prove here that the number of lattice points inside K can also
accept an upper bound of roughly the same size, in the special case where K is
an ellipsoid. Whether this is also true for all K unconditionally is an open
problem, but there is reasonable hope that the inductive approach used for
ellipsoids could be extended to all cases.Comment: 9 page
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