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.