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Paper IPM / M / 14979  


Abstract:  
Let (X,.) be a real normed space and let θ:(0,∞)→ (0,∞) be an increasing function such that t→ [(t)/(θ(t))] is nondecreasing on (0,∞). For such function, we introduce the notion of θangular distance α_{θ}[x,y], where x, y ∈ X\{0}, and show that X is an inner product space if and only if α_{θ}[x, y] ≤ 2 [(x− y)/(θx+θy)] for each x, y ∈ X\{0}. Then, in order to generalize the DunklWilliams constant of X, we introduce a new geometric constant C_{F}(X) for X wrt F, where F: (0, ∞)×(0, ∞)→ (0, ∞) is a given function, and obtain some characterizations of inner product spaces related to the constant C_{F}(X). Our results generalize and extend various known results in the literature.
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