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- W1973041430 abstract "Abstract In every inner product space H the Ptolemy inequality holds: the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides. In other words, ‖ x − y ‖ ‖ z − w ‖ ≤ ‖ x − z ‖ ‖ y − w ‖ + ‖ z − y ‖ ‖ x − w ‖ for any points w , x , y , z in H . It is known that for each normed space ( X , ‖ ⋅ ‖ ) , there exists a constant C such that for any w , x , y , z ∈ X , we have ‖ x − y ‖ ‖ z − w ‖ ≤ C ( ‖ x − z ‖ ‖ y − w ‖ + ‖ z − y ‖ ‖ x − w ‖ ) . The smallest such C is called the Ptolemy constant of X and is denoted by C P ( X ) . We study the relationships between this constant and the geometry of the space X , and hence with metric fixed point theory. In particular, we relate the Ptolemy constant C P to the Zbaganu constant C Z , and prove that if X is a Banach space with C Z ( X ) ( 1 + 3 ) / 2 , then X has (uniform) normal structure and therefore the fixed point property for nonexpansive mappings. We derive general lower and upper bounds for both C P and C Z , and calculate the precise values of these two constants for several normed spaces. We also present a number of conjectures and open problems." @default.
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- W1973041430 date "2010-06-01" @default.
- W1973041430 modified "2023-10-16" @default.
- W1973041430 title "The Ptolemy and Zbăganu constants of normed spaces" @default.
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- W1973041430 doi "https://doi.org/10.1016/j.na.2010.01.030" @default.
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