Antipodal pairs and the geometry of Banach spaces

Let X be a real Banach space. For 0 " 1 and x 2 X, define the quantities
(x, ") =
infkak=" max{ka + xk, ka − xk} and
X(") =
(") = supkxk=1
(x, "). We study the
basic properties of the above functions. For instance, they show that: (a)
(0) = 1
(x, ")

(") 1 + "; (b) for each fixed " the function X 7!
X(") is continuous with respect to the
Banach-Mazur distance; (c) if (") denotes the modulus of convexity of X, then (2"/
(")) <
1−1/
(") and (2/(1+")) < 1−("
(")/(1+")).
Wealso compute the above functions for certain classical Banach spaces. For instance,
they show that: (1) If X is one of c, l1, C[0, 1], or L1[0, 1], then
(") = 1+"; (2) if X = lp for
1 p <1, then
(") = (1+"p)1/p; (3) if X = Lp[0, 1] for 1 p < 2, then
(") = [((1−")p +
(1+")p)/2]1/p; (4) ifX is an AL or an AM-space, then for the Banach space L(X) of all bounded
operators on X one has
(") = 1+".