I shall consider the problem of minimizing the relative perimeter under a volume constraint in the interior of an unbounded convex body with arbitrary boundary. I shall give an example of a convex body whose isoperimetric profile is identically zero and give a characterization of the convex bodies with positive isoperimetric profile in terms of their asymptotic cylinders. I shall also show existence of isoperimetric regions in a generalized sense, and prove the concavity of the function I^{(n+1)/n}, where I is the isoperimetric profile and R^{n+1} is the ambient Euclidean space.
This is joint work in progress with Gian Paolo Leonardi and Efstratios Vernadakis.

Abstract