Consider a strictly convex surface $\Sigma$ in $\mathbb{R}^3$ homeomorphic to a sphere. When $p$ is a point not in a convex hull of $\Sigma$, then $\Sigma'$ is the boundary of convex hull of $p$ and $\Sigma $. Define $$C=\overline{ \Sigma' -\Sigma },\ D= \overline{\Sigma-\Sigma'}$$
When $|\ |_X$ is an intrinsic metric on the set, then let $x\in \partial C$. When $\gamma :[0,l]\rightarrow D$ is a unit speed shortest path in $D$ s.t. $$\gamma (0)=x,\ \gamma'(0)=\frac{ p-x}{|p-x|}$$ where $|\ |$ is a Euclidean metric in $\mathbb{R}^3$, then we let $l\in [0,|p-x|]$ to be the largest.
For any $y\in \partial C$, then $$|\gamma (l)-y|_{D}\leq |p-y|$$ This is true ?
