An illumination problem: optimal apex and optimal orientation for a cone of light.
Résumé
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 $\mathbb{Z} \subset \mathbb{R}^n$ and a unit vector $y \in \mathbb{R}^n$ so that the conic light beam $ C(z, y, s) := \{x \in \mathbb{R}^n : s \left \| x − z \right \| − <y, x − z> \leq 0 \} $ captures every $a_ i$ and, at the same time, it has a sharpness coefficient $s\in [0, 1]$ as large as possible.
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