Compactly uniformly convex spaces and property of Rolewicz

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