The Hausdorff distance between two subsets of a metric space is defined by , where are (open/closed does not matter) -neighborhoods of the sets. Informally: if no matter where you are in one set, you can jump into the other by traveling less than . For example, the distance between letters S and U is about the length of the longer green arrow.

Generally, one assumes the sets to be closed (to avoid zero distance) and bounded (to avoid infinite distance). But I will not; in this post I’m not really interested in verifying all the axioms of a metric.

The Gromov-Hausdorff distance is defined between metric spaces as follows: it is the infimum of taken over all isometric embeddings and into some metric space .

The infimum over all pairs of embeddings into all conceivable metric spaces does not sound like something you would want to compute in practice. Of course, the matter boils down to equipping the abstract union with pseudometrics that are compatible with the original metrics on and .

A more directly computable notion of distance (not necessarily a metric) can be given as follows: is the infimum of all for which there exist two maps and such that:

- for all
- for all
- for all
- for all

This is not as elegant as “infimize over all metric space”, but is more practical. For example, it is easy to check that the sequence of one-sheeted hyperboloids

converges to the cone .

Using cylindrical coordinates, define by and by , with an arbitrary choice of at the point . Now check the items one by one:

- is the identity map on
- fixes all points of except for those with . The latter are displaced by at most .
- follows from on
- follows from on

Note that the Gromov-Hausdorff convergence of manifolds is understood with respect to their intrinsic metrics. Although both and are naturally identified with subsets of , it would be a mistake to use the Hausdorff distance based on the Euclidean metric of . Even though this extrinsic metric is bi-Lipschitz equivalent to the intrinsic metric on both and , bi-Lipschitz equivalence is too coarse to preserve the GH convergence in general. In general, the intrinsic metric on a manifold cannot be realized as the extrinsic metric in any Euclidean space.

In the example of hyperboloids we are lucky to have a Gromov-Hausdorff convergent sequence of unbounded spaces. Normally, bounded sets are required, and boundedness is imposed either by an exhaustion argument, or by changing the metric (moving to the projective space). For instance, the parabolas do not converge to the line as unbounded subsets of , but they do converge as subsets of . In the process, the degree jumps down from to .

It seems that the Gromov-Hausdorff limit of algebraic varieties of degree at most is also an algebraic variety of degree at most (provided that we use the intrinsic metric; otherwise flat ellipses would converge to a line segment). If this is true, I’m sure there’s a proof somewhere but I never saw one.