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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 $\delta$-vector of the dual polytope $\mathcal{P}^\vee \subset\mathbb{R}^d$. In general, $\delta(\mathcal{P})=\delta(\mathcal{P}^\vee)$ doesnot hold. In this paper, we give a higher-dimensional construction of reflexivepolytope whose $\delta$-vector equals the $\delta$-vector of the dual polytope.In particular, we consider the case that the reflexive polytope and the dualpolytope are unimodularly equivalent.

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