*Accepted Paper*

**Inserted:** 23 jun 2023

**Last Updated:** 6 may 2024

**Journal:** Journal für die reine und angewandte Mathematik (Crelle's Journal)

**Year:** 2024

**Doi:** 10.1515/crelle-2024-0032

**Abstract:**

For every $d\ge 3$, we construct a noncompact smooth $d$-dimensional Riemannian manifold with strictly positive sectional curvature without isoperimetric sets for any volume below $1$. We construct a similar example also for the relative isoperimetric problem in (unbounded) convex sets in $\mathbb R^d$. The examples we construct have nondegenerate asymptotic cone. The dimensional constraint $d\ge 3$ is sharp. Our examples exhibit nonexistence of isoperimetric sets only for small volumes; indeed in nonnegatively curved spaces with nondegenerate asymptotic cones isoperimetric sets with large volumes always exist. This is the first instance of a nonnegatively curved space without isoperimetric sets.