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

In this paper, we investigate two properties concerning the unimodality of the $\delta$-vectors of lattice polytopes, which are log-concavity and alternatingly increasingness. For lattice polytopes $\mathcal{P}$ of dimension $d$, we prove that the dilated lattice polytopes $n\mathcal{P}$ have strictly log-concave and strictly alternatingly increasing $\delta$-vectors if $n > \max{s,d+1-s}$,where$s$isthedegreeofthe$\delta$-polynomialof$$\mathcal{P}$$\mathrm{.}Thebound$\max{s,d+1-s}$for$n$ is reasonable. We also provide several kinds of unimodal (or non-unimodal) $\delta$-vectors. Concretely, we give examples of lattice polytoeps 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, respectively.