The $\delta$-vectors of reflexive polytopes and of the dual polytopes

Let $\delta (\mathcal{P})$ be the $\delta$-vector of a reflexive polytope $\mathcal{P}\subset {\mathbb{R}}^d$ of dimension $d$ and $\delta($\mathcal{P}$ ^\vee)$the$\delta$-vectorofthedualpolytope$$\mathcal{P}$^\vee \subset $\mathbb{R}$^d$\mathrm{.}Ingeneral,$\delta($\mathcal{P}$)=\delta($\mathcal{P}$^\vee)$ does not hold. In this paper, we give a higher-dimensional construction of reflexive polytope whose $\delta$-vector equals the $\delta$-vector of the dual polytope. In particular, we consider the case that the reflexive polytope and the dual polytope are unimodularly equivalent.