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Suppose $M \cong \mathbb{Z}^n$ is a rank $n$ lattice, with dual lattice $N$. Suppose $\Delta$ is a full dimensional lattice polytope (i.e. convex hull of finite lattice points) in $M$. Then $\Delta$ is a reflexive polytope if and only if its dual polytope $\Delta^\vee =\{y \in M \otimes_\mathbb{Z} \mathbb{R} \mid \langle \Delta, y \rangle \geq -1\}$ is also a lattice polytope.

Let $\#(V(\Delta))$ denote the number of vertices of $\Delta$.

Is there any estimate on $\#(V(\Delta))$? Particularly, It would be great if we have bound like $$\max\{\#(V(\Delta)\mid \Delta ~{\rm{ is~ reflexive ~of dimensional~}}n\} \leq C n$$ for some constant $C$ not depending on $n$.

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A "cube" $[-1,1]^n$ has $2^n$ vertices and is reflexive.

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  • $\begingroup$ Indeed. But then excluding products of smaller reflexive polytopes, what are the known bounds ? $\endgroup$
    – F. C.
    Commented Sep 9, 2016 at 18:53
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Maybe I am too late, but Casagrande's "The Number of Vertices of a Fano Polytope" gives the bound $|V(P)|\leq 3n$ for $n$ the dimension of the polytope, provided that the polytope is simplicial.

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  • $\begingroup$ Thank you! Although I don't know why I asked that question:) $\endgroup$
    – Li Yutong
    Commented Sep 10 at 0:35

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