Английская Википедия: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.
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
- <math>(y, z)_{x} = \frac1{2} \big( d(x, y) + d(x, z) - d(y, z) \big).</math>
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 <math>d(x,y) = a + b, \ d(x,z) = a + c, \ d(y,z) = b + c</math>. Then the Gromov products are <math>(y,z)_x = a, \ (x,z)_y = b, \ (x,y)_z = c</math>. 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 Шаблон:Math, so that Шаблон:Math. 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]
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,
- <math>d(x, y) = (x, z)_{y} + (y, z)_{x},</math>
- <math>0 \leq (y, z)_{x} \leq \min \big\{ d(y, x), d(z, x) \big\},</math>
- <math>\big| (y, z)_{p} - (y, z)_{q} \big| \leq d(p, q),</math>
- <math>\big| (x, y)_{p} - (x, z)_{p} \big| \leq d(y, z).</math>
Points at infinity
Consider hyperbolic space Hn. Fix a base point p and let <math>x_\infty</math> and <math>y_\infty</math> be two distinct points at infinity. Then the limit
- <math>\liminf_{x \to x_\infty \atop y \to y_\infty} (x,y)_p</math>
exists and is finite, and therefore can be considered as a generalized Gromov product. It is actually given by the formula
- <math>(x_\infty, y_\infty)_{p} = \log \csc (\theta/2),</math>
where <math>\theta</math> is the angle between the geodesic rays <math>px_\infty</math> and <math>py_\infty</math>.[2]
δ-hyperbolic spaces and divergence of geodesics
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,
- <math>(x, z)_{p} \geq \min \big\{ (x, y)_{p}, (y, z)_{p} \big\} - \delta.</math>
In this case. 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).
Notes
References
