  We define the notion of isometric envelope of a subspace in a Banach space,
and relate it to a) the mean ergodic projection on the space of fixed points of
a semigroup of contractions, b) results on Korovkin sets from the 70's, and c)
extension properties of linear isometric embeddings. We use this concept to
address the recent conjecture that the Gurarij space and the spaces $L_p$, $p
\notin 2\mathbb N+4$ are the only separable Approximately Ultrahomogeneous
Banach spaces (a certain multidimensional transitivity of the action of the
linear isometry group). The similar conjecture for Fra\"iss\'e Banach spaces (a
strenghtening of the Approximately Homogeneous Property) is also considered. We
characterize the Hilbert space as the only separable reflexive space in which
any closed subspace coincides with its envelope. We compute some envelopes in
the case of Lebesgue spaces, showing that the reflexive $L_p$-spaces are the
only reflexive rearrangement invariant spaces on $[0,1]$ for which all
$1$-complemented subspaces are envelopes. We also identify the isometrically
unique "full" quotient space of $L_p$ by a Hilbertian subspace, for appropriate
values of $p$, as well as the associated topological group embedding of the
unitary group into the isometry group of $L_p$.
