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.

Let (X, d) be a metric space and let x, y, z ∈ X. 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 ${\displaystyle d(x,y)=a+b,\ d(x,z)=a+c,\ d(y,z)=b+c}$. Then the Gromov products are ${\displaystyle (y,z)_{x}=a,\ (x,z)_{y}=b,\ (x,y)_{z}=c}$. 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.

Consider hyperbolic space Hⁿ. Fix a base point p and let ${\displaystyle x_{\infty }}$ and ${\displaystyle y_{\infty }}$ be two distinct points at infinity. Then the limit

exists and is finite, and therefore can be considered as a generalized Gromov product. It is actually given by the formula

where ${\displaystyle \theta }$ is the angle between the geodesic rays ${\displaystyle px_{\infty }}$ and ${\displaystyle py_{\infty }}$.

The Gromov product can be used to define δ-hyperbolic spaces in the sense of Gromov.: (X, d) is said to be δ-hyperbolic if, for all p, x, y and z in X,