# Additive number theory and inequalities in Ehrhart theory

Alan Stapledon

Arxiv ID: 0904.3035•Last updated: 10/5/2021

We introduce a powerful connection between Ehrhart theory and additive number
theory, and use it to produce infinitely many new classes of inequalities
between the coefficients of the $h^*$-polynomial of a lattice polytope. This
greatly improves upon the three known classes of inequalities, which were
proved using techniques from commutative algebra and combinatorics. As an
application, we deduce all possible `balanced' inequalities between the
coefficients of the $h^*$-polynomial of a lattice polytope containing an
interior lattice point, in dimension at most 6.

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