We associate to a finite digraph D a lattice polytope P D whose vertices are the rows of the Laplacian matrix of D. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of P D equals the complexity of D, and P D contains the origin in its relative interior if and only if D is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, h * -polynomial, and integer decomposition property of P D in these cases. We extend Braun and Meyer's study of cycles by considering cycle digraphs. In this setting we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.

A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete enumeration of such equivalence classes for arbitrary constants d and K. The algorithm, which gives another proof of the finiteness result, is implemented for small values of K, up to dimension six. The resulting database contains and extends several existing ones, and has been used to correct mistakes in other classifications. When specialized to three-dimensional smooth polytopes, it extends previous classifications by Bogart et al., Lorenz and Lundman. Moreover, we give a structure theorem for smooth polytopes with few lattice points that proves that they have a quadratic triangulation and that we use, together with the classification, to describe smooth polytopes having small volume in arbitrary dimension. In dimension three we enumerate all the simplices having up to 11 interior lattice points and we use them to conjecture a set of sharp inequalities for the coefficients of the Ehrhart h˚-polynomials, unifying several existing conjectures. Finally, we extract and discuss minimal interesting examples from the classification, and we study the frequency of properties such as being spanning, very ample, IDP, and having a unimodular cover or triangulation. In particular, we find the smallest polytopes which are very ample but not IDP, and with a unimodular cover but without a unimodular triangulation.2010 Mathematics Subject Classification. 52B20 (Primary); 52B11 (Secondary).

In this paper, we show the existence of universal inequalities for the h * -vector of a lattice polytope P , that is, we show that there are relations among the coefficients of the h * -polynomial which are independent of both the dimension and the degree of P . More precisely, we prove that the coefficients h * 1 and h * 2 of the h * -vector (h * 0 , h * 1 , . . . , h * d ) of a lattice polytope of any degree satisfy Scott's inequality if h * 3 = 0.2010 Mathematics Subject Classification. Primary: 52B20; Secondary: 52B12.

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We give an upper bound on the volume vol(P * ) of a polytope P * dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree (−K X ) d of a d-dimensional toric Fano variety X with at worst canonical singularities.