The uniform structure of Banach spaces

We explore the existence of uniformly continuous sections for quotient maps. Using this approach we are able to give a number of new examples in the theory of the uniform structure of Banach spaces. We show for example that there are two non-isomorphic separable ${\mathcal L_1}$ -subspaces of ? ₁ which are uniformly homeomorphic. We also prove the existence of two coarsely homeomorphic Banach spaces (i.e. with Lipschitz isomorphic nets) which are not uniformly homeomorphic (answering a question of Johnson, Lindenstrauss and Schechtman). We construct a closed subspace of L ₁ whose unit ball is not an absolute uniform retract (answering a question of the author).