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}...</equationsource></equation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation>

We face the problem of strictly separating two sets of points by means of a sphere, considering the two cases where the center of the sphere is fixed or free, respectively. In particular, for the former we present a fast and simple solution algorithm, whereas for the latter one we use the...