Nearly optimal coverings of a sphere with generalized spherical segments

The subject of this study is the incomplete covering of a two-dimensional sphere by sets resulting from the intersection of this sphere with a cone whose vertex is inside the sphere. A numerical method is proposed for evaluating the criterial function of the covering, which is representable in the form of a multiple minimax. The problem of optimally choosing the axes for the cones defining the sets in the covering is examined. By exploiting symmetry considerations, this problem is reduced to a similar problem of small dimension, which can be numerically treated on modern computers. The reduction is performed by solving an auxiliary optimization problem. It is shown that the criterial function of this problem is Lipschitzian. The results of several numerical tests are presented. For possible computer implementations of the proposed methods, certain recommendations on parallelizing some numerical procedures are given.