Girths and Flat Banach Spaces

J. J. Schâffer [3] introduced an interesting parameter for normed linear spaces. It is termed girth and is the infimum of the lengths of all centrally symmetric simple closed rectifiable curves which lie in the boundary of the unit ball. More precisely, let X be a Banach space with norm denoted by || -|| and with dim X^2. K curve in X will be a rectifiable geometric curve defined by Busemann [l] as the equivalence class of curves (i.e., continuous functions from a compact interval of real numbers into the space X with the metric given by |j-1|) which have the same standard representation in terms of arc-length. Given a curve c we denote its length by X(c) ; and we denote by 7c(s), 0^5^X(c), its standard representation in terms of arclength. We say that 7c(0) and 7c(X(c)) are the initial and final points of c. A curve c is simple if 7C is injective. Following common usage, a curve c often stands for the common range of its parametrizations, which is a compact subset of X. Thus we say, for example, "xÇEc," or "the linear hull of c," or c lies in a subset A of X," etc. Let B denote the unit ball of X and S the unit sphere. As usual the inner metric 5 of S is defined for all x, y(ES by