NORMAL STRUCTURE AND THE ARC LENGTH IN BANACH SPACES

Abstract

Let X be a Banach space, $X_2 \subseteq X$ be a two dimensional subspace of $X$, and $S(X) = \{x \in X, ||x|| = 1\}$ be the unit sphere of $X$. The relationship between the normal structure and the arc length in X is studied. Let $R(X) = \mbox{inf} \{l(S(X_2)) - \gamma(X_22) : X_2 \subseteq X\}$, where $l(S(X_2))$ is the circumference of $S(X_2)$ and $\gamma(X_2) = \mbox{sup}\{2(||x + y|| + ||x - y||) : x; y \in S(X_2)\}$ is the least upper bound of the perimeters of the inscribed parallelogram of $S(X_2)$. The main result is that $R(X) \gt 0$ implies $X$ has the uniform normal structure.