An illumination problem with tradeoff between coverage of a dataset and aperture angle of a conic light beam

Let (Formula presented.) be a finite set in (Formula presented.). The illumination problem addressed in this work concerns the optimal location and orientation of a conic light beam (Formula presented.)The aperture angle (Formula presented.) of the conic light beam is a decreasing function of the sharpness coefficient (Formula presented.). The problem at hand is to select an apex z in a prescribed compact region (Formula presented.) and a unit vector (Formula presented.) so that the conic light beam R(z, y, s) fulfils two conflicting requirements: it captures as many points (Formula presented.) as possible and, at the same time, it has a sharpness coefficient s as large as possible.