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?



Simplicial polytope with regular cones
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.


