Gromov product

In mathematics, the Gromov product is a concept in the theory of metric spaces named after the mathematician Mikhail Gromov.

The Gromov product can also be used to define δ-hyperbolic metric spaces in the sense of Gromov.

Then the Gromov product of y and z at x, denoted (y, z)x, is defined by Given three points x, y, z in the metric space X, by the triangle inequality there exist non-negative numbers a, b, c such that

In the case that the points x, y, z are the outer nodes of a tripod then these Gromov products are the lengths of the edges.

In the hyperbolic, spherical or euclidean plane, the Gromov product (A, B)C equals the distance p between C and the point where the incircle of the geodesic triangle ABC touches the edge CB or CA.

Indeed from the diagram c = (a – p) + (b – p), so that p = (a + b – c)/2 = (A,B)C. Thus for any metric space, a geometric interpretation of (A, B)C is obtained by isometrically embedding (A, B, C) into the euclidean plane.

[1] Consider hyperbolic space Hn.

Fix a base point p and let

Then the limit exists and is finite, and therefore can be considered as a generalized Gromov product.

Gromov product measures how long geodesics remain close together.

Namely, if x, y and z are three points of a δ-hyperbolic metric space then the initial segments of length (y, z)x of geodesics from x to y and x to z are no further than 2δ apart (in the sense of the Hausdorff distance between closed sets).