Existence of Chebyshev centers, best 𝑛-nets and best compact approximants

Abstract

In this paper we investigate the existence and continuity of Chebyshev centers, best n n -nets and best compact sets. Some of our positive results were obtained using the concept of quasi-uniform convexity. Furthermore, several examples of nonexistence are given, e.g., a sublattice M M of C [ 0 , 1 ] C[0,\,1] , and a bounded subset B ⊂ M B \subset M is constructed which has no Chebyshev center, no best n n -net and not best compact set approximant.