Minkowski's convex body theorem

Let be a complete lattice and be a bounded convex subset symmetric about the origin. If

then contains a nonzero element of . #m/thm/geo Moreover, the constant cannot be any smaller. If is compact, then the same conclusion holds for the weaker hypothesis¹

Proof of first part

For the first part, suppose and consider the sublattice . By Covolume of a classical lattice, we have .

Let be a measurable fundamental domain for , and consider the map induced by the projection . Since

by the hypothesis, the Measure theoretic pigeonhole principle implies that is not injective, i.e. there exist distinct such that , whence . Let . By symmetry of , and by convexity

so is the required nonzero element.

Sharpness

It is already evident for and that the constant cannot be made smaller.

#state/develop | #lang/en | #SemBr

2022. Algebraic number theory course notes, ¶3.6, p. 62 ↩

Minkowski's convex body theorem

Footnotes