# Polar of integral polytope — confusing claim in Fulton

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.