Ehrhart polynomials with negative coefficients

It is shown that, for each $d \geq 4$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that each of the coefficients of $n, n^{2}, \ldots, n^{d-2}$ of its Ehrhart polynomial $i(\mathcal{P},n)$ is negative. Moreover, it is also shown that for each $d \geq 3$ and $1 \leq k \leq d-2$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that the coefficient of $n^k$ of the Ehrhart polynomial $i(\mathcal{P},n)$ of $\mathcal{P}$ is negative and all its remaining coefficients are positive. Finally, we consider all the possible sign patterns of the coefficients of the Ehrhart polynomials of low dimensional integral convex polytopes.