I am in particular uncertain about how the strong maximum principle is used in the argument below. Could someone please clarify and add more detailed explanations. Thanks
So assume we have a regular embedded curve $\gamma$ in $\mathbb{R}^2$ whose endpoints are on the boundary of a closed convex subset of $\mathbb{R}^2$.Furthermore assume it satisfies the curve shortening flow
$$\frac{\partial}{\partial t}\gamma=\frac{\partial^2}{\partial s^2}\gamma$$ If any part of the curve besides the endpoints touches the boundary of the convex subset, then by the strong maximum principle it follows that the flow makes the curve so that it no longer touches the boundary besides the endpoints for $t>0$.