  On a compact stratified space (X, g) there exists a metric of constant scalar
curvature in the conformal class of g, if the scalar curvature satisfies an
integrability condition and if the Yamabe constant of X is strictly smaller
than the local Yamabe constant , another conformal invariant introduced in the
recent work of K. Akutagawa, G. Carron and R. Mazzeo. Such invariant depends on
the local structure of X, in particular on the links, but its explicit value is
not known. We are going to show that if the links satisfy a Ricci positive
lower bound, then we can compute the local Yamabe constant. In order to achieve
this, we prove a lower bound for the spectrum of the Laplacian, by extending a
well-known theorem by Lichenrowicz, and a Sobolev inequality, inspired by a
result due to D. Bakry. Furthermore, we prove the existence of an Euclidean
isoperimetric inequality on particular stratified space, with one stratum of
codimension 2 and cone angle bigger than 2$\pi$.
