On a set of problems on covering of a bounded set (Q1825437)

The author describes a constructive approach to the search of admissible coverings of a polyhedron by translates of polyhedra. The domain \(S_ 0\) of the covering is a point set of the Euclidean space \(E^ k\), and \(S_ i\), \(i=1,...,n\), are the covering objects. The main attention is paid to the case when the objects \(S_ i\), \(i=0,...,n\), are polyhedra of the space \(E^ k\). A location of the covering objects when each point of the domain of the covering belongs to at least one object is called an admissible covering. The author can give such coverings for arbitrary polyhedra \(S_ i\). He shows the major stages of solutions of this problem and describes algorithms for auxiliary problems occuring at each stage by referring to the respective literature. A numerical example given at the end of the paper shows the acceptability of the approach given and, on the whole, shows difficulties of its realization.