Discretization and affine approximation in high dimensions
  Lower estimates are obtained for the macroscopic scale of affine approximability of vector-valued Lipschitz functions on finite dimensional normed spaces, completing the work of Bates, Johnson, Lindenstrass, Preiss and Schechtman. This yields a new approach to Bourgain's discretization theorem for superreflexive targets. 
Sean Li and Assaf Naor
math.FA math.MG