Upper bound on scaled Gromov-hyperbolic δ

The Gromov-hyperbolic δ or “fatness” of a hyperbolic geodesic triangle, defined to be the infimum of the perimeters of all inscribed triangles, is given an explicit analytical expression in term of the angle data of the triangle. By a hyperbolic extension of Fermat’s principle, the optimum inscribed triangle is easily constructed as the orthic triangle, that is, the triangle with its vertices at the feet of the altitudes of the original triangle. From the analytical expression of the optimum perimeter δ, a Tarski-Seidenberg computer algebra argument demonstrates that the δ, scaled by the diameter of the triangle, never exceeds 3/2 in a Riemannian manifold of constant nonpositive curvature. As probably the most important corollary, a finite metric geodesic space in which the ratio δ/diam is (strictly) bounded from above by 3/2 for all geodesic triangles exhibits the same metric properties as a negatively curved Riemannian manifold. The specific applications targeted here are those involving such very large but finite graphs as the Internet and the Protein Interaction Network. It is indeed argued that negative curvature is the precise mathematical formulation of their visually intuitive core concentric property.