Some inequalities concerning geometric constants of Banach spaces (Banach space theory and related topics)
Yasuji Takahashi, Mikio Kato · Institutional Repositories DataBase (IRDB) · 2011
Let $X$ be a real Banach space with $\dim X\geq 2$ .The closed unit ball and unit sphere of $X$ are denoted by $B_{X}$ and $S_{X}$ , respectively.We shall consider the following constants:Properties and relations concerning these constants have been studied by many authors.In particular the James constant $J(X)$ and the von Neumann-Jordan constant $C_{NJ}(X)$ have been most widely treated.Recall that a Banach space $X$ is uniformly non-square provided $J(X)<2$ or equivalently $C_{NJ}(X)<2$ .The constant $C_{NJ}'(X)$ may be considered as the unitary version of $C_{NJ}(X)$ ([2],[6]).The constant $C_{Z}(X)$ was introduced by $Zb\check{a}ganu[11]$ , who conjectured that $C_{NJ}(X)=$ $C_{Z}(X)$ for all Banach spaces $X$ , but in general these two constants are different.