Equivalent norms with the property $(\beta )$ of Rolewicz

We extend to the non separable setting many characterizations of the Banach spaces admitting an equivalent norm with the property $(\beta )$ of Rolewicz. These characterizations involve in particular the Szlenk index and asymptotically uniformly smooth or convex norms. This allows to extend easily to the non separable case some recent results from the non linear geometry of Banach spaces.