Volumes of Convex Lattice Polytopes and A Question of V. I. Arnold.

We show by a direct construction that there are at least exp $${\{cV^{(d-1)/(d+1)}\}}$$ convex lattice polytopes in $${\mathbb{R}^d}$$ of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. This is achieved by considering the family $${\mathcal{P}^{d}(r)}$$ (to be defined in the text) of convex lattice polytopes whose volumes are between 0 and r/ d!. Namely we prove that for $${P \in \mathcal{P}^{d}(r), d!}$$ vol P takes all possible integer values between cr and r where $${c > 0}$$ is a constant depending only on d. [ABSTRACT FROM AUTHOR]