A CONSTANT RELATED TO FIXED POINTS AND NORMAL STRUCTURE IN BANACH SPACES

Abstract

Let X be a Banach space and S_X = {x in X : ||x|| = 1} be the unit sphere of X.
The parameter

V_X^* (\epsilon) = sup { 1- \frac{||x+y||}{2} : x,y in S_X and <x-y, f> <= \varepsilon for some f in \triangledown_x },

where 0 <= \varepsilon <= 2 and \triangledown_x \subseteq S_X^* is the set of norm 1 supporting functionals f at x, is
introduced and investigated. The main result is that if V_X^* < \frac{ \varepsilon }{2} for some 0 < \varepsilon < 2, then
X is uniformly nonsquare and has a uniform normal structure.