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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?

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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.

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  • $\begingroup$ Do you have a brief argument for why OP's condition implies $P_0$ being reflexive? $\endgroup$
    – M. Winter
    Feb 24, 2022 at 12:57
  • $\begingroup$ 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. $\endgroup$
    – Francisco Santos
    Feb 25, 2022 at 14:06

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