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+".