On James and von Neumann–Jordan constants and sufficient conditions for the fixed point property

Abstract

In this paper, we prove that a Banach space X and its dual space X * have uniform normal structure if C NJ (X) < (1 + √ 3)/2. The García-Falset coefficient R(X) is estimated by the C NJ (X)-constant and the weak orthogonality coefficient introduced by B. Sims. Finally, we present an affirmative answer to a conjecture by L. Maligranda concerning the relation between the James and C NJ (X)-constants for a Banach space.

Wheeling around von Neumann–Jordan constant in Banach spaces

“…From (19) we obtain the estimate J(X * ) − J(X) ≤ 1 − J(X)/2 ≤ 1 − √ 2/2, and a similar one for J(X) − J(X * ). But probably this estimate is not sharp.…”

“…Some weaker inequalities have been proved in [16], while a solution of the problem has been claimed in [19], but with a doubtful proof. The next theorem answers the problem in the affirmative, giving an even sharper inequality.…”

“…Concerning J(X) no sharp inequality seems to be known. We can only deduce an estimate by using the inequalities (see [12, Theorem 1]) (19) 2J(X) − 2 ≤ J(X * ) ≤ J(X) 2 + 1.…”

Some Properties of Pythagorean Modulus

“…It is clear that 2 ≤ f ( ) ≤ 2(1 + 2 ) ≤ E( ) ≤ 2(1 + ) 2 . It is also worth noting that the first moduli E (X) has been proved to be very useful in the study of the well-known von Neumann-Jordan constant (see e.g., [3,4]). …”

On the Baronti constants of Orlicz function spaces

“…Note that J(X) is also called the James constant in [8] and [13]. A Banach space X is termed uniformly nonsquare in the sense of James if J(X) < 2.…”