Is there a standard construction of a metric on one-point compactification of a proper metric space?
Comments:
- A metric space is proper if all bounded closed sets are compact.
- Standard means found in literature.
From the answers and comments:
Here is a simplification of the construction given here1 (thanks to Jonas for ref).
Let $d$ be the original metric. Fix a point $p$ and set $h(x)=1/(1+d(p,x))$.
Then take the metric
$$\hat d(x,y)=\min\{d(x,y),\,h(x)+h(y)\},\ \ \ \ \hat d(\infty,x)=h(x).$$
A more complicated construction is given here2 (thanks to LK for ref), some call it "sphericalization".
One takes
$$\bar d(x,y)=d(x,y)\cdot h(x)\cdot h(y),\ \ \ \ \bar d(\infty,x)=h(x).$$
The function $\bar d$ does not satisfies triangle inequality, but one can show that there is a metric $\rho$ such that $\tfrac14\cdot \bar d\le \rho\le \bar d$.
1Mandelkern, Mark. “Metrization of the One-Point Compactification.” Proceedings of the American Mathematical Society, vol. 107, no. 4, American Mathematical Society, 1989, pp. 1111–15, https://doi.org/10.2307/2047675.
2Mario Bonk, Bruce Kleiner: Rigidity for Quasi-Mobius group actions
Best Answer
I will make this an answer although it is just a follow-up to the comment of LK
The recent names in this (but referring back to Bonk and Kleiner) are Stephen Buckley and David Herron, for proper spaces their one-point extension $\hat{X}$ is the one-point compactification, see pages 4 and 8 in
[PDF] METRIC SPACE INVERSIONS, QUASIHYPERBOLIC DISTANCE, AND UNIFORM SPACES File Format: PDF/Adobe Acrobat - Quick View by SM Buckley - 2008 - Cited by 3 - Related articles In a certain sense, inversion is dual to sphericalization. ... point compactification. All of the properties of dp mentioned above also hold for their con- ...
https://eprints.nuim.ie/1610/1/BuckleyMetricSpace.pdf