Let $z\in V^{\circ}$ and let $\mathcal{L}og$ be a branch of a holomorphic logarithm $(z_{1},\dots,z_{n+1})\mapsto(\log(z_{1}),\dots,\log(z_{n+1}))$ defined in a neighborhood of $z$. The point $z$ is critical for $\operatorname{Log}|_{V^{\circ}}$ iff $V^{\circ}$ and the orbit of the real torus $T^{n}$ are not transversal at $z$. But $\mathcal{L}og$ takes the tangent space to an orbit of $T^{n}$ to a translate of $i\mathbb{R}^{n+1}$ in $\mathbb{C}^{n+1}$.

Therefore, $z$ is critical iff $\mathcal{L}og(T_{z}V^{\circ})$ contains at least $n$ purely imaginary vectors which is, in turn, equivalent to $\gamma(z)\in\mathbb{R}P^{n}$. ∎