Ehrhart positivity of smooth polytopes with few facets

For each dimension $d$, we determine the minimum number of facets of a smooth $d$-dimensional lattice polytope whose Ehrhart polynomial has at least one negative coefficient. This minimum is six in dimension $d=3$, seven in dimension $d=4$, and $d+2$ in every dimension $d\ge 5$. The lower bounds follow from the classification of smooth lattice polytopes with at most $d+2$ facets, together with explicit formulas for Ehrhart polynomials in dimensions at most $4$. We also prove that a lattice polytope of dimension at least $3$ with a smooth vertex cannot be $2$-integral with respect to any ordered lattice basis.