Problems Related To Spherical Hulls Of Bounded Sets And Complete Sets

In 1911 Meissner introduced the concept of complete sets when he studied the sets of constant width in Euclidean spaces,a bounded set is called diametrically complete if it cannot be enlarged without increasing its diameter.About diametrically complete sets,many mathematicians have done a series of important work in general Banach spaces especially in finite dimensional Banach spaces around complete sets and its related properties,the diametric completion map and the problems related to complete sets,but there are still many unsolved problems.As we can see,spherical hull and properties of complete sets and the uniqueness of the completion of sets are closely linked and it plays an important role in the process of completion.We study the relation between the bounded sets and its wide spherical hull and tight spherical hull in this paper.The paper firstly reviews some fundamental properties of sets of constant width which is the origin of the concept of complete sets,properties of complete sets and research problems as well as corresponding results related to complete sets.Secondly,norms of some special normed linear spaces and some preliminaries about diametral points and balls are presented.Finally,we study the boundary structure of wide spherical hull and tight spherical hull of bounded sets in Banach spaces,diametral points of sets and its spherical hulls,and the distance from a bounded set to the boundary of its spherical hull.Moreover,a necessary and sufficient condition for the wide spherical hull of a bounded set be a ball is obtained.