Ice-Creams and Wedge Graphs

What is the minimum angle $\alpha >0$ such that given any set of $\alpha$-directional antennas (that is, antennas each of which can communicate along a wedge of angle $\alpha$), one can always assign a direction to each antenna such that the resulting communication graph is connected? Here two antennas are connected by an edge if and only if each lies in the wedge assigned to the other. This problem was recently presented by Carmi, Katz, Lotker, and Ros\'en \cite{CKLR10} who also found the minimum such $\alpha$ namely $\alpha=\frac{\pi}{3}$. In this paper we give a simple proof of this result. Moreover, we obtain a much stronger and optimal result (see Theorem \ref{theorem:main}) saying in particular that one can chose the directions of the antennas so that the communication graph has diameter $\le 4$. Our main tool is a surprisingly basic geometric lemma that is of independent interest. We show that for every compact convex set $S$ in the plane and every $0 < \alpha < \pi$, there exist a point $O$ and two supporting lines to $S$ passing through $O$ and touching $S$ at two \emph{single points} $X$ and $Y$, respectively, such that $|OX|=|OY|$ and the angle between the two lines is $\alpha$.