The Ptolemy constant of absolute normalized norms on ℝ2

Abstract

We determine and estimate the Ptolemy constant of absolute normalized norms on ℝ 2 by means of their corresponding continuous convex functions on [0, 1]. Moreover, the exact values were calculated in some concrete Banach spaces. 2000 Mathematics Subject Classification: 46B20.

On the Ptolemy constant of Lorentz sequence spaces and their duals

“…Moreover, we completely compute the Ptolemy constant of two-dimensional Lorentz sequence spaces d (2) (ω, q) (1 q < ∞) and their dual spaces d (2) (ω, q) * in the case of q = 1 and q 2. The new results which not only contain some previous results 9,20 , but also give some new results in some concrete Banach spaces.…”

“…We consider the Ptolemy constant C p (X ) of absolute normalized norms on 2 , the exact values of C p (X ) were also calculated in some classical Banach spaces 9 .…”

“…In order to get the exact value of C p (d (2) (ω, q) * ) in the case of q = 1, we need the following example 9 .…”

“…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] .…”

“…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 . However, to our best knowledge, there are no results about the exact value of Ptolemy constant C p (X ) in Lorentz sequence spaces d (2) (ω, q)(1 q < ∞ and q = 2) and their dual spaces.…”