Abstract. Let $(X,\|\cdot\|)$ be a normed space, $$\eqalign{C_J(X) &=\sup\{\| x+y\|\wedge\| x-y\|:x,y\in S(X)\};\cr C_S(X) &=\inf\{\| x+y\|\vee\| x-y\|:x,y\in S(X)\}.}$$ In this paper we will show that if $X$ is a real normed space with $\dim(X)>1$, then $C_J(X)C_S(X)=2$. For some classical Banach spaces we get that $$\eqalign{C_J(X) &=\sup\{\| x+y\|:\| x+y\|=\| x-y\|, x,y\in S(X)\};\cr C_S(X) &=\inf\{\| x+y\|:\| x+y\|=\| x-y\|, x,y\in S(X)\}.}$$ We also give the expression of $C_J(X)$, $C_S(X)$ in two classes of Orlicz spaces, which involves the result in $L^p$.

AMS Subject Classification (1991): 46B30

Keyword(s): James nonsquare constant, Schäffer nonsquare constant, Orlicz space, Uniformly nonsquare

Received January 4, 1994 and in revised form June 20, 1994. (Registered under 5595/2009.)