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Lattice points : Convex region symm. about the origin

  1. Jul 30, 2007 #1
    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 ?
     
  2. jcsd
  3. Aug 1, 2007 #2

    Gokul43201

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