一致凸Banach空间的一个新的特征性质

A New Characteristic Property of Uniformly Convex Banach Spaces

给出了Banach空间一致凸的一个新的充要条件:设λ,μ∈(0,1),λ+μ=1,f:R+R+是单调递增且可微的严格凸函数,X是Banach空间,则X是一致凸的当且仅当对任意ε>0,存在δ>0,使得当‖x‖≤1,‖x-y‖≥ε时,有f(‖λx+μy‖)<λf(‖x‖)+μf(‖y‖)

A new sufficient and necessary condition is given for uniformly convex Banach spaces. The main result is the following theorem. Theorem Suppose that λ,μ∈（0,1）,λ＋μ=1,f：R^＋→R^＋ is a increasing, convex function and X is a Banach space. Then X is uniformly convex if and only if for everye ε〉 0 there existsδ〉0 such that f（｜｜λx＋μy｜｜）〈λf（｜｜x｜｜）＋μf（｜｜y｜｜）-δ for all ｜｜x｜｜≤1and y∈x satisfying ｜｜x-Y｜｜≥ε