Best possible lower bounds on the coefficients of Ehrhart polynomials

For an integral convex polytope P⊂Rd, we recall LP(n)=|nP∩Zd| the Ehrhart polynomial of P. Let gr(P) be the rth coefficients of LP(n) for r=0,…,d. Martin Henk and Makoto Tagami gave lower bounds on the coefficients gr(P) in terms of the volume of P. They proved that these bounds are best possible for r∈{1,2,d−2}. We show that these bounds are also optimal for r=3 and d−r even and we give a new best possible bound for r=d−3.