  It is well known that isoperimetric regions in a smooth compact
$(n+1)$-manifold are smooth, up to a closed set of codimension at most $6$. In
this note, we first construct an $8$-dimensional compact smooth manifold whose
unique isoperimetric region with half volume that of the manifold exhibits two
isolated singularities. And then, for $n\geq 7$, using Smale's construction of
singular homological area minimizers for higher dimensions, we construct a
Riemannian manifold such that the unique isoperimetric region of half volume,
with singular set the submanifold $\S^{n-7}$.
