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| Paper IPM / M / 14979 |
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| Abstract: | |
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Let (X,||.||) be a real normed space and let θ:(0,∞)→ (0,∞) be an increasing function such that t→ [(t)/(θ(t))] is non-decreasing on (0,∞). For such function, we introduce the notion of θ-angular distance αθ[x,y], where x, y ∈ X\{0}, and show that X is an inner product space if and only if αθ[x, y] ≤ 2 [(||x− y||)/(θ||x||+θ||y||)] for each x, y ∈ X\{0}. Then, in order to generalize the Dunkl-Williams constant of X, we introduce a new geometric constant CF(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 CF(X). Our results generalize and extend various known results in the literature.
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