Normal structure and the arc length in Banach spaces

Let X be a Banach space, X-2 subset of or equal to X be a two dimensional subspace of X, and S(X) = {x is an element of X, //x// = 1} be the unit sphere of X. The relationship between the normal structure and the arc length in X is studied. Let R(X) = inf{l(S(X-2)) - r(X-2) : X-2 subset of or equal to X}, where l(S(X-2)) is the circumference of S(X-2) and r(X-2) = sup{2(//x + y// + //x - y//) : x,y is an element of S(X-2)} is the least upper bound of the perimeters of the inscribed parallelogram of S(X-2). The main result is that R(X) > 0 implies X has the uniform normal structure.