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)}, 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.