View source for Flatness theorem

''(in the geometry of numbers)''

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f1101301.png" /> be a closed bounded [[Convex set|convex set]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f1101302.png" /> of non-zero volume. If the width of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f1101303.png" /> is at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f1101304.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f1101305.png" /> contains an element of the integer lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f1101306.png" />.

Here, the width of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f1101307.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f1101308.png" />) is the minimum width of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f1101309.png" /> along any non-zero integer vector. The width of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f11013010.png" /> along a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f11013011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f11013012.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f11013013.png" /></td> </tr></table>

The width of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f11013015.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f11013016.png" /> is greater or equal than the geometric width of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f11013017.png" />, which is the minimum width of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f11013018.png" /> along all unit-length vectors.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110130/f11013019.png" /> 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|Frobenius problem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Hastad,   "Dual vectors and lower bounds for the nearest lattice point problem"  ''Combinatorica'' , '''8'''  (1988)  pp. 75–81</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR></table>