ON BANAS-HAJNOSZ-WEDRYCHOWICZ TYPE MODULUS OF CONVEXITY AND FIXED POINT PROPERTY

Abstract

LetXbeaBanachspacewiththeunitsphereSX ={x∈X:∥x∥=1}. In this paper, inspired by Banas et. al., in [1], the new parameter SY_X (ε) = sup{⟨x − y, fx⟩ : x,y ∈ S_X, ∥x+y∥/2 ≥ 1−ε for some f_x ∈ ∇_x}, where ∇_x ⊆ S_X^∗ is the norm 1 supporting functionals at x, is introduced. Several properties of this parameter are investigated. The main result are that if SY_X (t) < 2, for some t ∈ (0, 1] then X is uniformly non-square; and if SY_X (ε) < 1+2ε for some 0 < ε < 1/2, then both X and X^∗ have uniform normal structure. In particular, if ε_U = lim_{ε→0} SY_X (ε) < 1, then X is uniformly non-square and both X and X^∗ have uniform normal structure. We have an example to show this condition is the best possible.