Isoperimetric inequalities in unbounded convex bodies

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