The discrete Legendre transform enables us to reproduce Batyrev duality [5]. Let $\Delta\subseteq M_{\mathbb{R}}$ be a reflexive polytope, $\nabla\subseteq N_{\mathbb{R}}$ the polar dual, and assume $0\in\Delta$ is the unique interior point. We then obtain two toric degenerations given by the equations

in $\mathbb{P}_{\Delta}\times\mathbb{A}^{1}$ and $\mathbb{P}_{\nabla}\times\mathbb{A}^{1}$ respectively, with $s_{m}$ ($s_{n}$) the section of $\mathcal{O}_{\mathbb{P}_{\Delta}}(1)$ corresponding to $m$ (the section of $\mathcal{O}_{\mathbb{P}_{\nabla}}(1)$ corresponding to $n$). It is easy to check that the dual intersection complexes of these two degenerations are given as follows. For the first degeneration, $B=\partial\nabla$ with polyhedral decomposition given by the proper faces of $\nabla$. The fan structure at each vertex $v$ is given by projection $U_{v}\hookrightarrow N_{\mathbb{R}}\rightarrow N_{\mathbb{R}}/\mathbb{R}v$. For the second degeneration, one uses $\Delta$ instead of $\nabla$. One can then check that if one polarizes the two degenerations using $\mathcal{O}_{\mathbb{P}_{\Delta}}(1)$ and $\mathcal{O}_{\mathbb{P}_{\nabla}}(1)$ respectively, then the corresponding triples $(B,\mathscr{P},\varphi)$ are Legendre dual. Thus Batyrev duality is a special case of this general approach to a mirror construction.

For a much more general construction which works for the Batyrev-Borisov construction [6] of mirrors of complete intersection Calabi-Yaus in toric varieties, see [22].