  In this paper we consider the problem of minimizing the relative perimeter
under a volume constraint in the interior of a convex body, i.e., a compact
convex set in Euclidean space with interior points. We shall not impose any
regularity assumption on the boundary of the convex set. Amongst other results,
we shall prove the equivalence between Hausdorff and Lipschitz convergence, the
continuity of the isoperimetric profile with respect to the Hausdorff
distance,and the convergence in Hausdorff distance of sequences of
isoperimetric regions and their free boundaries. We shall also describe the
behavior of the isoperimetric profile for small volume, and the behavior of
isoperimetric regions for small volume.
