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Examples of IDP lattice polytopes with non-log-concave h^*-vector
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Examples of IDP lattice polytopes with non-log-concave h^*-vector
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Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a $k$th dilation is a sum of $k$ lattice points in the polytope. It is a long-standing conjecture whether the numerator of the Ehrhart series of an IDP polytope, called the $h^*$-polynomial, has a unimodal coefficient vector. In this preliminary report on research in progress we present examples showing that $h^*$-vectors of IDP polytopes do not have to be log-concave. This answers a question of Luis Ferroni and Akihiro Higashitani. As this is an ongoing project, this paper will be updated with more details and examples in the near future.
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Unimodular polytopes and column number bounds on polytopal totally unimodular matrices via Seymour's decomposition theorem
A sharp upper bound is established on distinct columns of unit-sum polytopal totally unimodular matrices and on vertices of unimodular polytopes.
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