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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 another one by a lattice preserving affine transformation. This is achieved byconsidering the family $\mathcal{P}^d(r)$ (to be defined in the text) of convexlattice polytopes whose volumes are between $0$ and $r^d/d!$. Namely we provethat for $P \in \mathcal{P}^d(r)$, $d!\mathrm{vol\;} P$ takes all possibleinteger values between $cr^{d-1}$ and $r^d$ where $c>0$ is a constant dependingonly on $d$.

Volumes of convex lattice polytopes and a question of V. I. Arnold