Comparison of -convexity, -convexity and other geometrical properties

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 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).