## Taiwanese Journal of Mathematics

### NORMAL STRUCTURE AND THE ARC LENGTH IN BANACH SPACES

Ji Gao

#### 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.

#### Article information

Source
Taiwanese J. Math., Volume 5, Number 2 (2001), 353-366.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407342

Digital Object Identifier
doi:10.11650/twjm/1500407342

Mathematical Reviews number (MathSciNet)
MR1832173

Zentralblatt MATH identifier
0984.46011

#### Citation

Gao, Ji. NORMAL STRUCTURE AND THE ARC LENGTH IN BANACH SPACES. Taiwanese J. Math. 5 (2001), no. 2, 353--366. doi:10.11650/twjm/1500407342. https://projecteuclid.org/euclid.twjm/1500407342