The Ptolemy and Zbăganu constants of normed spaces

The Ptolemy constant of absolute normalized norms on ℝ2

“…It has been shown that these constants are very useful in geometric theory of Banach spaces, which enable us to classify several important concepts of Banach spaces such as uniformly non-squareness and uniform normal structure [3][4][5][6][7][8]. On the other hand, calculation of the constant for some concrete spaces is also of some interest [5,6,9].…”

“…By S X and B X we denote the unit sphere and the unit ball of a Banach space X, respectively. The notion of the Ptolemy constant of Banach spaces was introduced in [10] and recently it has been studied by Llorens-Fuster in [9]. Definition 1.1 For a normed space (X, ||.||) the real number C p (X) := sup x − y z x − z y + z − y x : x, y, z ∈ X\{0}, x = y = z = x is called the Ptolemy constant of (X, ||.||).…”

On the Ptolemy constant of Lorentz sequence spaces and their duals

“…Recently, we give a simple method to determine the Ptolemy constant C p (X ) of absolute normalized norms on 2 which are complementary to the results of Llorens-Fuster, Mazcunan-Navarro and Reich 7 . Moreover, the exact values of the Ptolemy constant C p (X ) were calculated in some classical Banach spaces, such as the space p , Cesàro space ces (2) p , and Lorentz sequence spaces d (2) (ω, 2) 9,18,19 .…”

“…From the above definition, we can get that C Z (X ) C p (X ). However, there is no relationship between the Ptolemy constant C p (X ) and von Neumann-Jordan constants C N J (X ) 7 .…”

“…Alternatively, one can do this with the help of some geometric constants. Some geometric constants for a Banach space X have been investigated in the literature, such as von Neumann-Jordan constants C N J (X ) [1][2][3] , Zbǎganu constant C Z (X ) [4][5][6] , Ptolemy constant C p (X ) [7][8][9] . These constants are important due to its strong connection with some useful geometric properties, such as uniformly nonsquareness and uniform normal structure [10][11][12] .…”