Generic orbits and type isolation in the Gurarij space

We study the question of when the space of embeddings of a separable Banach space $E$ into the separable Gurarij space $\mathbf G$ admits a generic orbit under the action of the linear isometry group of $\mathbf G$. The question is recast in model-theoretic terms, namely type isolation and the existence of prime models. We characterise isolated types over $E$ using tools from convex analysis. We show that if the set of isolated types over $E$ is dense, then a dense $G\_\delta$ orbit exists, and otherwise all orbits are meagre. We then study some (families of) examples with respect to this dichotomy. We also point out that the class of Gurarij spaces is the class of models of an $\aleph\_0$-categorical theory with quantifier elimination, and calculate the density character of the space of types over $E$, answering a question of Avil{\'e}s et al.