Flat -vectors and their Ehrhart polynomials

We call the delta-vector of an integral convex polytope of dimension d flat if the delta-vector is of the form (1, 0,..., 0, a,..., a, 0,..., 0), where a = 1. In this paper, we give the complete characterization of possible flat delta-vectors. Moreover, for an integral convex polytopeP. RN of dimension d, we let i(P, n) = /nP n ZN / and i*(P, n) = / n(P P) n ZN /. By this characterization, we show that for any d = 1 and for any k, l = 0 with k+ l = delta-1, there exist integral convex polytopes P and Q of dimension delta such that (i) For t = 1, ..., k, we have i(P, t) = i(Q, t), (ii) For t = 1, ..., l, we have i*(P, t) = i*(Q, t), and (iii) i(P, k+ 1) l = i(Q, k+ 1) and i*(P, l + 1) l = i*(Q, l + 1).