Using the Gromov product in inappropriate ways

Question

The Gromov product $(x,y)_z=1/2(d(z,x)+d(z,y)-d(y,x)$ is used in Gromov hyperbolic groups to measure how long two rays stay together or how thin a triangle is. In particular, if $(x,y)_z=n$ in a $\delta$-hyperbolic space, then the two rays from z to x and z to y stay within $\delta$ of each other for a length of at least $n-\delta$.

Similarly, if $(x,y)_p\geq \min\{(x,z)_p+(y,z)_p\}-\delta$ for all $x,y,z,p$ and some $\delta$, then triangles are $\delta$-thin.

So what happens when you try doing these things in Euclidean space? Is their a geometric interpretation of the Gromov product in terms of two rays or triangles?

Answer 1

Of course, rays in $\mathbb R^n$ typically do not stay within bounded distance from each other, so that interpretation isn't available. The inequality with three products and fixed $\delta$ no longer speaks of the shape of triangles as much as of their size, since the Gromov product is multiplied by $\lambda$ under the scaling $x\mapsto \lambda x$. (Such scaling, i.e., a map that multiplies all distances by the same factor, is not available in hyperbolic spaces.) It is equivalent to an upper bound on the radius of inscribed circle; as such, it holds for some triangles and fails for others.

But the quantity $d(z,x)+d(z,y)−d(y,x)$ is useful in Euclidean space as well, specifically for the Traveling Salesman problem: cover a given set $E\subset \mathbb R^n$ with a curve of finite (preferably small) length. If $E$ is finite, the curve is a graph. The quantity $d(z,x)+d(z,y)−d(y,x)$ is the price you pay for connecting $z$ to the graph by replacing $[x,y]$ with the edges leading to $z$. The farthest insertion algorithm works to minimize this price by making $[x,y]$ long.