1) Since the matrix $(g_{ij})$ defined by the metric $g_{Y}$ is positive, the function $K$ is convex. In particular, there is locally well-defined Legendre transform of $K$. This fact will be used later, when we will discuss the duality of Monge-Ampère manifolds.

2) It seems plausible that in the case when all $X_{q}$ are simply-connected, and $h^{k,0}(X_{q})=0$ for $0<k<n$, the metric space $\overline{Y}$ is a homological sphere of dimension $n$. In all examples it is in fact homeomorphic to $S^{n}$.