  We obtain a full resolution result for minimizers in the exterior
isoperimetric problem with respect to a compact obstacle in the large volume
regime $v\to\infty$. This is achieved by the study of a Plateau-type problem
with free boundary (both on the compact obstacle and at infinity) which is used
to identify the first obstacle-dependent term (called {\it isoperimetric
residue}) in the energy expansion, as $v\to\infty$, of the exterior
isoperimetric problem. A crucial tool in the analysis of isoperimetric residues
is a new mesoscale flatness criterion for hypersurfaces with bounded mean
curvature, which we obtain as a development of ideas originating in the theory
of minimal surfaces with isolated singularities.
