  We study the way in which the Euclidean subspaces of a Banach space fit
together, somewhat in the spirit of the Ka\v{s}in decomposition. The main tool
that we introduce is an estimate regarding the convex hull of a convex body in
John's position with a Euclidean ball of a given radius, which leads to a new
and simplified proof of the randomized isomorphic Dvoretzky theorem. Our
results also include a characterization of spaces with nontrivial cotype in
terms of arrangements of Euclidean subspaces.
