Maximal volume enclosed by plates and proof of the chessboard conjecture

The following conjecture of Fejes T6th is proved: The density of a lattice of convex bodies in [w” is at least i, provided that each connected component of the complementary set is bounded. The cases of equality are also determined. For any polytope P in [w” one can define a convexification P* (see Section 1) and it is proved that Vol P S Vol P*. It is also shown that if P and P’ are two convex polytopes with facets 4 and FT, resp., and ~{Vol,_,~~~F;u}~~{Vol,_,F~/F~~v} holds for every UE[W”, then Vol P’IVol P < 2”‘(“-‘). So me related questions are also considered.