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Flatness theorem

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(in the geometry of numbers)

Let be a closed bounded convex set in of non-zero volume. If the width of is at least , then contains an element of the integer lattice .

Here, the width of (with respect to ) is the minimum width of along any non-zero integer vector. The width of along a vector in is

The width of with respect to is greater or equal than the geometric width of , which is the minimum width of along all unit-length vectors.

If is a rational polyhedron, i.e. is defined by a system of linear inequalities with rational coefficients, then the "non-zero volume condition" in the flatness theorem can be dispensed with. The flatness theorem finds application in, e.g., the Frobenius problem.

References

[a1] J. Hastad, "Dual vectors and lower bounds for the nearest lattice point problem" Combinatorica , 8 (1988) pp. 75–81
[a2] J. Lagarias, H.W. Lenstra, C.P. Schnorr, "Korkine–Zolotarev bases and successive minima of a lattice and its reciprocal lattice" Combinatorica , 10 (1990) pp. 333–348
How to Cite This Entry:
Flatness theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flatness_theorem&oldid=16956
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article