Coarse and uniform embeddings

In these notes, we study the relation between uniform and coarse embeddings between Banach spaces as well as uniform and coarse equivalences. In order to understand these relations better, we look at the problem of when a coarse embedding can be assumed to be also topological. Among other results, we show that if a Banach space X uniformly embeds into a minimal Banach space Y, then X simultaneously coarsely and uniformly embeds into Y, and if a Banach space X coarsely embeds into a minimal Banach space Y, then X simultaneously coarsely and homeomorphically embeds into Y by a map with uniformly continuous inverse.