Method for constructing optimal dark coverings

The problem of constructing metric $\varepsilon$-nets and corresponding coverings by balls for compact sets with a probability measure is considered. In the case of sets having metrically significant parts with a small measure (dark sets), methods for constructing $\varepsilon$-nets are combined with the deep holes method in a unified approach. According to this approach, a constructed metric net is supplemented with its deep hole (the most distant element of the set) until the required accuracy is achieved. An existing implementation of the method for a metric set with a given probability measure is based on a pure global search for deep holes. To construct dark coverings, the method is implemented on the basis of a random multistart. For the resulting nets, the logarithm of the number of their elements is shown to be close to $\varepsilon$-entropy, which means that they are optimal. Techniques for estimating the reliability and completeness of constructed $(\varepsilon, \delta)$-coverings in the sense of C.E. Shannon are described. The methods under consideration can be used to construct coverings of implicitly given sets with a measure defined on the preimage and to recover compact supports of multidimensional random variables with an unknown distribution law.