A pythagorean approach in Banach spaces

Abstract

Let X be a Banach space and let S(X) ={ x ∈ X, � x �= 1}be the unit sphere of X. Parameters E(X) = sup{α(x), x ∈ S(X)}, e(X) = inf {α(x), x ∈ S(X)}, F(X) = sup{β(x), x ∈ S(X)} ,a ndf (X) = inf {β(x), x ∈ S(X)} ,w hereα(x) = sup{� x + y� 2 + � x − y� 2 , y ∈ S(X)} ,a ndβ(x) = inf {� x+ y� 2 + � x − y� 2 , y ∈ S(X)} are introduced and studied. The values of these parameters in the lp spaces and function spaces Lp[0,1] are estimated. Among the other results, we proved that a Banach space X with E(X) 2i s uniform nonsquare; and a Banach space X with E(X) 32/9 has uniform normal structure.