  We prove that any proper, geodesic metric space whose Dehn function grows
asymptotically like the Euclidean one has asymptotic cones which are
non-positively curved in the sense of Alexandrov, thus are ${\rm CAT}(0)$. This
is new already in the setting of Riemannian manifolds and establishes in
particular the borderline case of a result about the sharp isoperimetric
constant which implies Gromov hyperbolicity. Our result moreover provides a
large scale analog of a recent result of Lytchak and the author which
characterizes proper ${\rm CAT}(0)$ in terms of the growth of the Dehn function
at all scales. We finally obtain a generalization of this result of Lytchak and
the author. Namely, we show that if the Dehn function of a proper, geodesic
metric space is sufficiently close to the Euclidean Dehn function up to some
scale then the space is not far (in a suitable sense) from being ${\rm CAT}(0)$
up to that scale.
