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

Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\{a_i:i\in I\}$$</EquationSource> </InlineEquation> be a finite set in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mathbb R ^n$$</EquationSource> </InlineEquation>. The illumination problem addressed in this work is about selecting an apex <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$z$$</EquationSource> </InlineEquation> in a prescribed set <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$Z\subseteq \mathbb R ^n$$</EquationSource> </InlineEquation> and a unit vector <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$y\in \mathbb R ^n$$</EquationSource> </InlineEquation> so that the conic light beam <Equation ID="Equ55"> <EquationSource Format="TEX">$$\begin{aligned} C(z,y,s):= \{x \in \mathbb R ^n : s\,\Vert x-z\Vert - \langle y, x-z\rangle \le 0\} \end{aligned}$$</EquationSource> </Equation>captures every <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$a_i$$</EquationSource> </InlineEquation> and, at the same time, it has a sharpness coefficient <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$ s\in [0,1]$$</EquationSource> </InlineEquation> as large as possible. Copyright Springer Science+Business Media New York 2014