Lower bounds on the coefficients of Ehrhart polynomials

We present lower bounds for the coefficients of Ehrhart polynomials of convex lattice polytopes in terms of their volume. Concerning the coefficients of the Ehrhart series of a lattice polytope we show that Hibi’s lower bound is not true for lattice polytopes without interior lattice points. The counterexample is based on a formula of the Ehrhart series of the join of two lattice polytope. We also present a formula for calculating the Ehrhart series of integral dilates of a polytope.

💡 Research Summary

The paper investigates two fundamental aspects of Ehrhart theory for convex lattice polytopes: (i) lower bounds for the coefficients of Ehrhart polynomials expressed in terms of the polytope’s Euclidean volume, and (ii) the validity of Hibi’s lower bound for the coefficients of the Ehrhart series when the polytope has no interior lattice points. In addition, the authors derive a compact formula for the Ehrhart series of integral dilates of a polytope.
The first main result establishes that for a d‑dimensional lattice polytope (P) with Ehrhart polynomial
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