Flatness theorem
(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 |
Flatness theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flatness_theorem&oldid=16956