I read a comment about the $\delta$-thin tetrahedra property of a space. It basically means, that if you choose any four points in this space, connect them by geodesics, and fill each triangle with a minimal filling (?). Then any point on one of the faces has distance at most $\delta$ to a point on one of the other faces.

For example, I think, that $\mathbb{R}^2,\mathbb{H}^n$ have this property and $\mathbb{R}^3$ has not. So it seems to be a nice generalization of a $\delta$-hyperbolic space. Has somebody a reference for this "thin tetrahedra property"?

Furthermore I was wondering which properties of Gromov-hyperbolic spaces might have analogues in this setting:

1) Is there some analogue of the Gromov product?

2) Which spaces can be the asympotic cone of a space with the $\delta$-thin tetrahedra property?

3) Which spaces have the $0$-thin tetrahedra property ?