In a Minkowski space with unit ball S, a set D(\(\lambda)\) is diametrically complete if \(D(\lambda)=\cap \{d+\lambda S:\) \(d\in D(\lambda)\}\). In this paper a method is given for generating such sets D(\(\lambda)\) which contain an arbitrary set X. This technique is then employed to find necessary and sufficient conditions for a set A to lie on the boundary of some such D(\(\lambda)\), and bounds are given on the acceptable values of \(\lambda\).