Comparison of PP-convexity, OO-convexity and other geometrical properties

We study some properties of OO-convex and PP-convex spaces and give a new proof of the P(n)P(n)-convexity of ℓp sums of P(n)P(n)-convex spaces. Further we show using Ramsey’s theorem that the space EβEβ is PP-convex. We define some moduli that characterize OO-convexity and PP-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 PP-convexity from several geometrical conditions known to imply the fixed point property (FPP).