Maximal metrically regular sets

Metrically regular sets form an interesting subclass of all subsets of an arbitrary finite discrete metric space $M$. Let us denote $\widehat{S}$ the set of points which are at maximal possible distance from the subset $S$ of the space $M$. Then $S$ is called metrically regular, if the set of vectors which are at maximal possible distance from $\widehat{S}$ coincides with $S$. The problem of investigating metrically regular sets appears when studying bent functions, set of which is metrically regular in the Boolean cube with the Hamming metric. In this paper the method of obtaining metrically regular sets from an arbitrary subset of the metric space is presented. Smallest metrically regular sets in the Boolean cube are described, and it is proven that metrically regular sets of maximal cardinality in the Boolean cube have covering radius $1$ and are complements of minimal covering codes of radius $1$. Lower bound on the sum of cardinalities of a pair of metrically regular sets, each being metric complement of the other, is given.