Given a simple closed Jordan (i.e. continuous) curve $\gamma:[a,b]\to\mathbb{R}^2,\ \gamma([a,b])=C_\gamma$, how can I prove that $D_\gamma$ (the set of all interior points of $\gamma$ toghether with the boundary $C_\gamma$) admits exactly two supporting lines in each direction?
A support line for $D_\gamma$ is a line that contain at least one point from $C_\gamma$ but no interior point of $D_\gamma$.
I didn't find any references for that convex geometry question..., but I saw that this property is used in the geometry of constant width curves.
The orthogonal projection of $D_\gamma$ onto any line is a connected, compact set with nonempty interior: thus, a closed interval of positive length. The endpoints of this interval correspond to supporting lines. The interior points of the interval correspond to lines $L$ such that $D_\gamma\setminus L$ is disconnected: since $D_\gamma$ is a topological disk, the set $D_\gamma\cap L$ must contain some of its interior points.