Editing Minkowski's theorem (section)

==Proof==
The following argument proves Minkowski's theorem for the specific case of <math>L = \mathbb{Z}^2.</math>

'''Proof of the <math display = "inline"> \mathbb{Z}^2 </math> case:''' Consider the map
:<math>f: S \to \mathbb{R}^2/2L, \qquad (x,y) \mapsto (x \bmod 2, y \bmod 2)</math>
Intuitively, this map cuts the plane into 2 by 2 squares, then stacks the squares on top of each other. Clearly {{math|''f''&thinsp;(''S'')}} has area less than or equal to 4, because this set lies within a 2 by 2 square. Assume for a [[proof by contradiction|contradiction]] that {{math|''f''}} could be [[injective]], which means the pieces of {{math|''S''}} cut out by the squares stack up in a non-overlapping way. Because {{math|''f''}} is locally area-preserving, this non-overlapping property would make it area-preserving for all of {{math|''S''}}, so the area of {{math|''f''&thinsp;(''S'')}} would be the same as that of {{math|''S''}}, which is greater than 4. That is not the case, so the assumption must be false: {{math|''f''}} is not injective, meaning that there exist at least two distinct points {{math|''p''<sub>1</sub>, ''p''<sub>2</sub>}} in {{math|''S''}} that are mapped by {{math|''f''}} to the same point: {{math|''f''&thinsp;(''p''<sub>1</sub>) {{=}} ''f''&thinsp;(''p''<sub>2</sub>)}}.

Because of the way {{math|''f''}} was defined, the only way that {{math|''f''&thinsp;(''p''<sub>1</sub>)}} can equal {{math|''f''&thinsp;(''p''<sub>2</sub>)}} is for {{math|''p''<sub>2</sub>}}
to equal {{math|''p''<sub>1</sub> + (2''i'', 2''j'')}} for some integers {{math|''i''}} and {{math|''j''}}, not both zero.
That is, the coordinates of the two points differ by two [[parity (mathematics)|even]] integers. 
Since {{math|''S''}} is symmetric about the origin, {{math|−''p''<sub>1</sub>}} is also a point in {{math|''S''}}. Since {{math|''S''}} is convex, the line segment between {{math|−''p''<sub>1</sub>}} and {{math|''p''<sub>2</sub>}} lies entirely in {{math|''S''}}, and in particular the midpoint of that segment lies in {{math|''S''}}. In other words,
:<math>\tfrac{1}{2}\left(-p_1 + p_2\right) = \tfrac{1}{2}\left(-p_1 + p_1 + (2i, 2j)\right) = (i, j)</math>
is a point in {{math|''S''}}. This point {{math|(''i'', ''j'')}} is an integer point, and is not the origin since {{math|''i''}} and {{math|''j''}} are not both zero.
Therefore, {{math|''S''}} contains a nonzero integer point.

'''Remarks:'''  
* The argument above proves the theorem that any set of volume <math display = "inline"> >\!\det(L)</math> contains two distinct points that differ by a lattice vector. This is a special case of [[Blichfeldt's theorem]].<ref>{{cite book
 | last1 = Olds | first1 = C. D. | author1-link = Carl D. Olds
 | last2 = Lax | first2 = Anneli | author2-link = Anneli Cahn Lax
 | last3 = Davidoff | first3 = Giuliana P. | author3-link = Giuliana Davidoff 
 | contribution = Chapter 9: A new principle in the geometry of numbers
 | isbn = 0-88385-643-3
 | mr = 1817689
 | page = 120
 | publisher = Mathematical Association of America, Washington, DC
 | series = Anneli Lax New Mathematical Library
 | title = The Geometry of Numbers
 | title-link = The Geometry of Numbers
 | volume = 41
 | year = 2000}}</ref>
* The argument above highlights that the term <math display = "inline">2^n \det(L)</math> is the covolume of the lattice <math display = "inline">2L</math>.
* To obtain a proof for general lattices, it suffices to prove Minkowski's theorem only for <math display = "inline">\mathbb{Z}^n</math>; this is because every full-rank lattice can be written as <math display = "inline">B\mathbb{Z}^n</math> for some [[linear transformation]] <math display = "inline">B</math>, and the properties of being convex and symmetric about the origin are preserved by linear transformations, while the covolume of <math display = "inline">B\mathbb{Z}^n</math> is <math display = "inline">|\!\det(B)|</math> and volume of a body scales by exactly <math display = "inline">\frac{1}{\det(B)}</math> under an application of <math display = "inline">B^{-1}</math>.