A Pythagorean Approach in Banach Spaces

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)}, and f (X) = inf{β(x), x ∈ S(X)}, where α(x) = sup{‖x + y‖2 + ‖x − y‖2, y ∈ S(X)}, and β(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) < 8, or f (X) > 2 is uniform nonsquare; and a Banach space X with E(X) < 5, or f (X) > 32/9 has uniform normal structure.