Unimodality of $$\delta $$ δ -vectors of lattice polytopes and two related properties

Abstract

We investigate two properties related to the unimodality of $$\delta $$ -vectors of lattice polytopes, which are log-concavity and alternatingly increasingness. For lattice polytopes P of dimension d, we prove that the dilated lattice polytopes nP have strictly log-concave and strictly alternatingly increasing $$\delta $$ -vectors if , where s is the degree of the $$\delta $$ -polynomial of P. We also provide several kinds of unimodal (or non-unimodal) $$\delta $$ -vectors. Concretely, we give examples of lattice simplices whose $$\delta $$ -vectors are not unimodal, unimodal but neither log-concave nor alternatingly increasing, alternatingly increasing but not log-concave, and log-concave but not alternatingly increasing.