# Simplicial polytope with regular cones

Let $$P$$ be a convex simplicial polytope in $$\mathbb{R}^n$$. Can we find a convex simplicial polytope $$P_0$$ in $$\mathbb{R}^n$$ combinatorially equivalent to $$P$$, satisfying the following condition: The vertices of $$P_0$$ are lattice points and for every facet $$F$$ of $$P_0$$ its vertices $$v_1,\dots,v_n$$ span $$\mathbb{Z}^n$$?

In other words, is a simplicial polytope combinatorially equivalent to a simplicial polytope such that the vertices of the facets and the origin form a unimodular simplex?

The conditions you pose on $$P_0$$ imply that it is a reflexive polytope. (That is, a lattice polytope with the origin in its interior and such that its polar dual is also a lattice polytope).
There are finitely many reflexive polytopes in each dimension (modulo $$GL(\mathbb Z,n)$$), which implies that the answer to your question is negative.
For example, in dimension two you can easily construct $$P_0$$ for a triangle, quadrilateral, pentagon, and hexagon, but there exists no reflexive heptagon.
• Do you have a brief argument for why OP's condition implies $P_0$ being reflexive? Feb 24 at 12:57
• As the OP says, the condition implies that "the vertices of [each of] the facets and the origin form a unimodular simplex". This implies that the facet can be written as $ax \le 1$ for an integer vector $a$. Thus, the polar polytope has integer vertices, namely the $a$'s coming from the facets. Together with the fact that $P_0$ has integer vertices this says that $P_0$ is reflexive. Feb 25 at 14:06