In the proposition on page 24 of Fulton's book "Toric Varieties," the following is claimed (slightly paraphrasing and omitting some other results):

Let $K$ be a polytope, and $K^{\circ} = \{ u : \langle u,v \rangle \geq - 1 \}$ be its polar. If the vertices of $K$ lie in a lattice $E$, then the vertices of $K^{\circ}$ lie in the dual lattice.

This statement doesn't appear to be true, for example, we can take $E = \mathbb{Z}$, and $K = \{2\}$, and then $K^{\circ} = [-1/2, \infty)$. (We found this in the literature: Exercise 2.2.1d in Cox, Little and Schenck asks the reader to construct an example of an integral polytope whose dual is not integral, using the formula $(r K )^{\circ} = (1/r) K^{\circ}$.)

The claim is also not true if we reinterpret it as being about the situation when $E$ is the lattice generated by the vertices of $K$; For example, a polygon in $\mathbb{R}^2$ containing containing $(-1,2),(0,2),(1,1),(1,0)$ along its boundary generates the lattice $\mathbb{Z}^2$, but the face defined by $\{(-1,2),(0,2)\}$ gives dual vertex $(1/2)y$, which is not in the dual lattice of $\mathbb{Z}^2$.

I'm confused about what the correct interpretation of this statement is.


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