An illumination problem: optimal apex and optimal orientation for a cone of light

Let $\{a_i:i\in I\}$ be a finite set in $\mathbb R ^n$ . The illumination problem addressed in this work is about selecting an apex $z$ in a prescribed set $Z\subseteq \mathbb R ^n$ and a unit vector $y\in \mathbb R ^n$ so that the conic light beam $$\begin{aligned} C(z,y,s):= \{x \in \mathbb R ^n : s\,\Vert x-z\Vert - \langle y, x-z\rangle \le 0\} \end{aligned}$$ captures every $a_i$ and, at the same time, it has a sharpness coefficient $ s\in 0,1]$ as large as possible.