Absolute norms on C-n1

In this paper, we study the absolute normalized norms on C-n. We show that the set of all absolute normalized norms on C-n is in one-to-one correspond ence with the set Psi (n) of all continuous convex functions on Delta (n) w ith some suitable conditions, where Delta (n) = {(s(1),...,s(n-1)) is an el ement of Rn-1; s(1) + s(2) + ... + s(n-1) less than or equal to 1, s(i) gre ater than or equal to 0 (1 less than or equal to i less than or equal to n - 1)}. As some applications, we show that an absolute norm on C-n is strict ly convex if and only if the corresponding function psi on Delta (n) is str ictly convex. Further, we consider the von Neumann-Jordan constant of C-n w ith an absolute normalized norm. (C) 2000 Academic Press.