Minimal ball-coverings in Banach spaces and their application

作者：Cheng Lixin^*; Cheng Qingjin; Shi Huihua

来源：Studia Mathematica, 2009, 192(1): 15-27.

DOI：10.4064/sm192-1-2

• ball-covering
• reflexivity
• smoothness
• differentiability
• Banach space

By a ball-covering B of a Banach space X, we mean a collection of open balls off the origin in X and whose union contains the unit sphere of X; a ball-covering, B is called minimal if its cardinality B(#) is smallest among all ball-coverings of X. This article, through establishing a characterization for existence of a ball-covering in Banach spaces, shows that for every n is an element of N with k <= n there exists an n-dimensional space admitting a minimal ball-covering of n k balls. As an application, we give a new characterization of superreflexive spaces in terms of ball-coverings. Finally, we show that every infinite-dimensional Banach space admits an equivalent norm such that there is an infinite-dimensional quotient space possessing a countable ball-covering.

• 出版日期2009
• 单位厦门大学