Ehrhart polynomials with negative coefficients

It is shown that, for each $d\ge 4$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that each of the coefficients of $n,n^2,\dots ,n^{d-2}$ of its Ehrhart polynomial $i(\mathcal{P},n)$ is negative. Moreover, it is also shown that for each $d\ge 3$ and $1 \leq k \leq d-2$,thereexistsanintegralconvexpolytope$$\mathcal{P}$$ofdimension$d$suchthatthecoefficientof$n^k$oftheEhrhartpolynomial$i($\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.