Asymptotic geometry of Banach spaces and uniform quotient maps
  Recently, Lima and Randrianarivony pointed out the role of the property $(\beta)$ of Rolewicz in nonlinear quotient problems, and answered a ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. In the present paper, we prove that the modulus of asymptotic uniform smoothness of the range space of a uniform quotient map can be compared with the modulus of $(\beta)$ of the domain space. We also provide conditions under which this comparison can be improved. 
S. J. Dilworth, Denka Kutzarova, G. Lancien, N. L. Randrianarivony
math.FA math.MG