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We find lower bounds for the set of Lipschitz constants of a given Lipschitzian map, defined on the closed unit ball of a Hilbert space, with respect to any renorming. We introduce a class of maps, defined in the closed unit ball of l 2 , which contains the classical fixed point free maps due to Goebel–Kirk–Thelle, Baillon, and P.K. Lin. We show that for any map of this class its uniform Lipschitz constant with respect to any renorming of l 2 is never strictly less than π 2 .

In this paper we prove some norm inequalities in the classical James sequence space J , and use them to prove that orthogonal convexity is a stable property in J .

We study some properties of O -convex and P -convex spaces and give a new proof of the P ( n ) -convexity of l p sums of P ( n ) -convex spaces. Further we show using Ramsey’s theorem that the space E β is P -convex. We define some moduli that characterize O -convexity and P -convexity and compare them to a modulus defined by Garcia Falset. This enables us to establish some fixed point theorems. By means of several examples, we separate P -convexity from several geometrical conditions known to imply the fixed point property (FPP).

In this paper we introduce two new classes of generalized nonexpansive mapping and we study both the existence of fixed points and their asymptotic behavior.

In this paper we show that the well-known Monch fixed point theorem for non-self mappings remains valid if we replace the Leray–Schauder boundary condition by the interior condition. As a consequence, we obtain a partial generalization of Petryshyn's result for nonexpansive mappings.

We consider the triangles with vertices x, −x and y where x,y are points on the unit sphere of a normed space. Using the geometric means of the variable lengths of the sides of these triangles, we define two geometric constants for Banach spaces. These constants are closely related to the modulus of convexity of the space under consideration, and they seem to represent a useful tool to estimate the exact values of the James and Jordan–von Neumann constants of some Banach spaces.

In this note we give a sufficient condition for the fixed point property for nonexpansive mappings in Banach spaces, which we call the AMCP. This property is more general than normal structure. and the space c 0 , James space, as well as every Banach space endowed with a suitable Schauder basis also have the AMCP. On the other hand, a renorming of the Tsirelson space and also 1 ∞ lack the AMCP.

In this paper we exhibit some connections between the Dunkl–Williams constant and some other well-known constants and notions. We establish bounds for the Dunkl–Williams constant that explain and quantify a characterization of uniformly nonsquare Banach spaces in terms of the Dunkl–Williams constant given by M. Baronti and P.L. Papini. We also study the relationship between Dunkl–Williams constant, the fixed point property for nonexpansive mappings and normal structure.

Let A be the set of all equivalent norms on l 1 which satisfy the FPP. We prove that A contains rays. In fact, every renorming in l 1 which verifies condition (⁎) in Theorem 2.1 is the starting point of a (closed or open) ray composed by equivalent norms on l 1 with the FPP. The standard norm ‖ ⋅ ‖ 1 or P.K. Linʼs norm defined in Lin (2008) [12] are examples of such norms. Moreover, we study some topological properties of the set A with respect to some equivalent metrics defined on the set of all norms on l 1 equivalent to ‖ ⋅ ‖ 1 .