Lattice points : Convex region symm. about the origin

[LeoYard]

Let R be a convex region symmetrical about the origin with area greater than 4. Show that R must contain a lattice point different from the origin.

This is the 2-D case of Minkowski's theorem, right ?

How about the n-dimensional version ?

The n-dimensional version is : Given a convex region R symmetrical to the origin in the n-dimensional space.

How to show that if R has volume greater than 2^n, then R contains a lattice point different from the origin ?

 

[Gokul43201]

Both look like weaker versions of the (2D and generalized) Minkowski convex body theorem.