I'm trying to understand a lemma from this paper. In my understanding, it says that
Since $M$ is closed, there exists $\alpha>0$ and a map $\phi:M\to \mathbb{R}^n$ (with some property that I am not familiar with) such that $\phi (B_M(x,\alpha))=B_{\mathbb{R}^n}(\mathbf{0},1) $. What is this map?
For any closed curve $\gamma$ on the boundary of $B_{\mathbb{R}^n}(\mathbf{0},1)$, the area of $D'=\mathbf{0}\circledast \gamma$, the cone over $\gamma$ with respect to the origin, is less or equal than the square of the length of $\gamma$. Is there any way to prove this using some integral?