We consider the problem of minimizing the relative perimeter under
a volume constraint in an unbounded convex body C ? Rn, 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.
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