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.
Definition 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
( y , z ) x = 1 2 ( d ( x , y ) + d ( x , z ) − d ( y , z ) ) . {\displaystyle (y,z)_{x}={\frac {1}{2}}{\big (}d(x,y)+d(x,z)-d(y,z){\big )}.}
Motivation
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 d ( x , y ) = a + b , d ( x , z ) = a + c , d ( y , z ) = b + c {\displaystyle d(x,y)=a+b,\ d(x,z)=a+c,\ d(y,z)=b+c} . Then the Gromov products are ( y , z ) x = a , ( x , z ) y = b , ( x , y ) z = c {\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.
Properties The Gromov product is symmetric: (y, z)x = (z, y)x. The Gromov product degenerates at the endpoints: (y, z)y = (y, z)z = 0. For any points p, q, x, y and z,
d ( x , y ) = ( x , z ) y + ( y , z ) x , {\displaystyle d(x,y)=(x,z)_{y}+(y,z)_{x},}
0 ≤ ( y , z ) x ≤ min { d ( y , x ) , d ( z , x ) } , {\displaystyle 0\leq (y,z)_{x}\leq \min {\big \{}d(y,x),d(z,x){\big \}},}
| ( y , z ) p − ( y , z ) q | ≤ d ( p , q ) , {\displaystyle {\big |}(y,z)_{p}-(y,z)_{q}{\big |}\leq d(p,q),}
| ( x , y ) p − ( x , z ) p | ≤ d ( y , z ) . {\displaystyle {\big |}(x,y)_{p}-(x,z)_{p}{\big |}\leq d(y,z).}
Points at infinity Consider hyperbolic space Hn. Fix a base point p and let x ∞ {\displaystyle x_{\infty }} and y ∞ {\displaystyle y_{\infty }} be two distinct points at infinity. Then the limit
lim inf x → x ∞ y → y ∞ ( x , y ) p {\displaystyle \liminf _{x\to x_{\infty } \atop y\to y_{\infty }}(x,y)_{p}}
exists and is finite, and therefore can be considered as a generalized Gromov product. It is actually given by the formula
( x ∞ , y ∞ ) p = log csc ( θ / 2 ) , {\displaystyle (x_{\infty },y_{\infty })_{p}=\log \csc(\theta /2),}
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