This is an announcement for the paper "A geometry characteristic for Banach space with $c^1$-norm" by Jipu Ma.

Abstract: Let $E$ be a Banach space with the $c^1$-norm $|\cdot|$ in $ E \backslash {0}$ and $S(E)={e\in E: |e|=1}.$ In this paper, a geometry characteristic for $E$ is presented by using a geometrical construct of $S(E).$ That is, the following theorem holds : the norm of $E$ is of $c^1$ in $ E \backslash {0}$ if and only if $S(E)$ is a $c^1$-submanifold of $E,$ with ${\rm codim}S(E)=1.$ The theorem is very clear, however, its proof is non-trivial, which shows an intrinsic connection between the continuous differentiability of the norm $|\cdot|$ in $ E \backslash {0}$ and differential structure of $S(E).$

Archive classification: math.FA

Mathematics Subject Classification: 54Exx, 46Txx, 58B20

Submitted from: huangql@yzu.edu.cn

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1109.6823

or

http://arXiv.org/abs/1109.6823