Let $V\subset{\mathbb{C}}{\mathbb{P}}^{n+1}\supset(\mathbb{C}^{*})^{n+1}$ be a projective hypersurface of degree $d$ not passing through the points $[1:0:\dots:0],\dots,[0:\dots:0:1]$. Then ${V}^{\circ}=V\cap(\mathbb{C}^{*})^{n+1}$ is given by a polynomial $f$ whose Newton polyhedron is

$$\Delta_{d}=\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}\ |\ 0\leq x_{j},\sum\limits_{j}x_{j}\leq d\}.$$

Vice versa, $\mathbb{C}T_{\Delta}={\mathbb{C}}{\mathbb{P}}^{n+1}$ and the closure of ${V}^{\circ}$ in ${\mathbb{C}}{\mathbb{P}}^{n+1}$ is $V$.