A Geometric Property of Banach Spaces Related to the Fixed Point Property

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 c0, 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.