On the structure of the set of equivalent norms on with the fixed point property

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 .