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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 ?

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I don't understand what a minimal filling is. Can you, please, give a reference where you saw this? –  Victor Protsak Jun 16 '10 at 7:01
    
Where did you see it? My feeling that "minimal" can not be the right choice; maybe barycentric (?). Also, you probably want to work only with Hadamard spaces (i.e. curvature $\le0$ and s.c.), is it correct? –  Anton Petrunin Jun 16 '10 at 10:08
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I found the comment in math.uni.wroc.pl/~elsner/papers/flats.pdf at the end of the first section. This is in the context of systolic complexes. Every geodesic triangle has a simplicial filling and then one can consider a filling with the minimal number of triangles. But it sounded somehow like there might be a definition for other kinds of metric spaces e.g. CAT(0) spaces. But I wasn'T sure, so I wanted to ask, whether someone has already heard of this. –  HenrikRüping Jun 16 '10 at 15:54

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