Compactly uniformly convex spaces and property (β)(β) of Rolewicz

We study different facets of property (β)(β) of Rolewicz. We remark that the notions of compact uniform convexity and property (β)(β) are isometrically equivalent and present new examples of spaces with that property. An observation is made that the property (β)(β) can be formulated in terms of graphs and an estimate of the (β)(β)-modulus is also given.