isoperimetric problems on Alexandrov spaces

Question

For an Alexandrov space M with curvature bounded from below, the isoperimetric profile $v \to I_M(v)$ defined for every $v\in (0,V(M))$ (the volume of M might be infinite), is given by $$ I_M(v)=inf\{A(\partial D): V(D)=v, D \subset \subset M\}, $$ where D varies over relatively compact open subset of M.

Then given any $v\in (0,V(M))$, is there a subset D with $A(\partial D)=I_M(v)$?

And are there any discription and regularity theorem for D and $\partial D$?

If we haven't results for general n, then what about the two-dimensional case?

Answer 1

The existence follows from theory of currents the same way as for Riemannian manifolds.

As far as I know, there are no regularity results for $\partial D$. But look at the proof of Levy--Gromov isoperimetric inequality in Alexandrov space [Theorem C in my "Harmonic functions on Alexandrov spaces..."] it does not use regularity.