Comparison of P-convexity, O-convexity and other geometrical properties

AbstractWe study some properties of O-convex and P-convex spaces and give a new proof of the P(n)-convexity of ℓp sums of P(n)-convex spaces. Further we show using Ramsey’s theorem that the space Eβ is P-convex. We define some moduli that characterize O-convexity and P-convexity and compare them to a modulus defined by García 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)