An upper bound for min-max angle of polygons

Let $S$ be a set of points in the plane, $CH$ be the convex hull of $S$, $\wp(S)$ be the set of all simple polygons crossing $S$, $\gamma_P$ be the maximum angle of polygon $P \in \wp(S)$ and $\theta =min_{P\in\wp(S)} \gamma_P$. In this paper, we prove that $\theta\leq 2\pi-\frac{2\pi}{r.m}$ such that $m$ and $r$ are the number of edges and internal points of $CH$, respectively. We also introduce an innovative polynomial time algorithm to construct a polygon with the said upper bound on its angles. Constructing a simple polygon with angular constraint on a given set of points in the plane is highly applicable to the fields of robotics, path planning, image processing, GIS, etc. Moreover, we improve our upper bound on $\theta$ and prove that this is tight