Abstract. A new constant $C(X)$ for a Banach space $X$ is introduced and it is proved that $X$ has the weak Banach--Saks property whenever $C(X)< 2$. Morover, it is shown that the Kottman constant $D(X)< 2$ implies that $X$ has the Banach--Saks property whenever $X$ is a Köthe sequence space. Nakano sequence spaces with the Banach--Saks property are charaterized. It is also proved that Cesàro sequence space $\mathop{\rm ces} _p$ has Banach--Saks type $p$.

AMS Subject Classification (1991): 46B20, 46E30

Keyword(s): Köthe Sequence Space, Cesàro Sequence Space, Packing Constant, Banach--Saks Property, Nakano Sequence Space, Weak Banach--Saks Property

Received January 26, 1998 and in revised form September 29, 1998. (Registered under 2680/2009.)