Around one decade ago several authors studied the parameter so defined: $$A_2(X) = \sup \{ \frac{||x + y|| + ||x -y||}{2}: x,y \in S_X\}$$ where $S_X$ denotes the unit sphere of the real Banach space $X$. In this paper we consider the new family of parameters that generalize $A_2(X)$: $$A_{2,p}(X) = \sup \big\{ \frac{||x+y|| + ||x-y||}{2}: \;\,x,y \in X, \;\,|| \, (||x||,||y||) \,||_p \leq 2 ^\frac{1}{p} \big\}$$ In this way, $A_{2,\infty}(X)$ is nothing else than $A_2(X)$ and we show how some interesting properties of real Banach spaces can be characterizing by using our new constants.
Triangles and parameters in normed spaces
BARONTI, MARCO;
2010-01-01
Abstract
Around one decade ago several authors studied the parameter so defined: $$A_2(X) = \sup \{ \frac{||x + y|| + ||x -y||}{2}: x,y \in S_X\}$$ where $S_X$ denotes the unit sphere of the real Banach space $X$. In this paper we consider the new family of parameters that generalize $A_2(X)$: $$A_{2,p}(X) = \sup \big\{ \frac{||x+y|| + ||x-y||}{2}: \;\,x,y \in X, \;\,|| \, (||x||,||y||) \,||_p \leq 2 ^\frac{1}{p} \big\}$$ In this way, $A_{2,\infty}(X)$ is nothing else than $A_2(X)$ and we show how some interesting properties of real Banach spaces can be characterizing by using our new constants.
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