Some characterizations of the complex projective space via Ehrhart polynomials

Let P-lambda Sigma n be the Ehrhart polynomial associated to an integral multiple lambda of the standard simplex Sigma(n) subset of R-n. In this paper, we prove that if (M, L) is an n-dimensional polarized toric manifold with associated Delzant polytope Delta and Ehrhart polynomial P-Delta such that P-Delta = P-lambda Sigma n, for some lambda is an element of Z(+), then (M, L) congruent to (CPn, O(lambda)) (where O (1) is the hyperplane bundle on CPn) in the following three cases: (1) arbitrary n and lambda = 1, (2) n = 2 and lambda = 3 and (3) lambda = n + 1 under the assumption that the polarization L is asymptotically Chow semistable.