  We show that, the solutions of the isoperimetric problem for small volumes
are $C^{2,\alpha}$-close to small spheres. On the way, we define a class of
submanifolds called pseudo balls, defined by an equation weaker than constancy
of mean curvature. We show that in a neighborhood of each point of a compact
riemannian manifold, there is a unique family concentric pseudo balls which
contains all the pseudo balls $C^{2,\alpha}$-close to small spheres. This
allows us to reduce the isoperimetric problem for small volumes to a
variational problem in finite dimension.
