Coverings of sets with restrictions on the arrangement of circles

Coverings of sets and domains by a system of circles whose centers have restrictions on their arrangement are considered. From the sensorrelated perspective, this corresponds to the sensor coverage problem, where the network sensors control objects or a region of space, but are located outside the control area. Formalizing the problem, we arrive at the problem of discrete geometry to determine the optimal number of circles, their sizes and locations, which provide the minimum coverage density of a given set. New results for the optimal outer covering of a circle, a square and a regular triangle are presented. The study of these models opens a possibility of building a sensor network with a minimal energy consumption.