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The notion of strictly outward minimising hull is investigated for open setsof finite perimeter sitting inside a complete noncompact Riemannian manifold.Under natural geometric assumptions on the ambient manifold, the strictlyoutward minimising hull $\Omega^*$ of a set $\Omega$ is characterised as amaximal volume solution of the least area problem with obstacle, where theobstacle is the set itself. In the case where $\Omega$ has $\mathscr{C}^{1,\alpha}$-boundary, the area of $\partial \Omega^*$ is recovered as the limit ofthe $p$-capacities of $\Omega$, as $p \to 1^+$. Finally, building on theexistence of strictly outward minimising exhaustions, a sharp isoperimetricinequality is deduced on complete noncompact manifolds with nonnegative Riccicurvature, provided $3 \leq n \leq 7$.

Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds