proof of Minkowski’s theorem
Proof. Let be any fundamental parallelepiped. Then obviously
(where means disjoint union) and thus
Now, note that
(draw a picture!) and thus, since measure is preserved by translation,
so that if all the are disjoint, we have
which is a contradiction. Thus there must exist and such that
Thus since is convex and centrally symmetric, and certainly , so we have found a nonzero element of .
Let be an arbitrary lattice in and let be the area of a fundamental parallelepiped. Any compact convex region symmetrical about the origin with contains a point of the lattice other than the origin.
Since is compact and thus closed, .
|Title||proof of Minkowski’s theorem|
|Date of creation||2013-03-22 17:53:41|
|Last modified on||2013-03-22 17:53:41|
|Last modified by||rm50 (10146)|