  We consider the problem of minimizing the relative perimeter under a volume
constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without
assuming any further regularity on the boundary of $C$. Motivated by an example
of an unbounded convex body with null isoperimetric profile, we introduce the
concept of unbounded convex body with uniform geometry. We then provide a handy
characterization of the uniform geometry property and, by exploiting the notion
of asymptotic cylinder of $C$, we prove existence of isoperimetric regions in a
generalized sense. By an approximation argument we show the strict concavity of
the isoperimetric profile and, consequently, the connectedness of generalized
isoperimetric regions. We also focus on the cases of small as well as of large
volumes; in particular we show existence of isoperimetric regions with
sufficiently large volumes, for special classes of unbounded convex bodies. We
finally address some questions about isoperimetric rigidity and analyze the
asymptotic behavior of the isoperimetric profile in connection with the notion
of isoperimetric dimension.
