Complete sets in normed linear spaces

Abstract

A bounded subset of a (finite or infinite dimensional) normed linear space is said to be complete (or diametrically complete) if it cannot be enlarged without increasing its diameter. Any bounded subset A of a normed linear space is contained in a complete set having the same diameter, which is called a completion of A. We survey characterizations, basic properties, facts about structure of the interior and boundary, and the asymmetry of complete sets. Different methods to obtain completions of bounded sets are presented. Moreover, the structure of the space of complete sets endowed with the Hausdorff metric and relations of this set family to related set families and notions are discussed. For example, we mean here sets of constant width, balls, reduced sets, sets of constant diameter, and sets of constant radius.