the Gromov-Hausdorff distance
/ GROH-mof HOWS-dorf /
How do you say that two shapes are 'almost the same shape,' even when they live in different worlds and are not subsets of any common space? Picture a fine triangular mesh and the smooth surface it approximates: they are different metric spaces, but as the mesh refines they should converge. The Gromov-Hausdorff distance is a way to measure how far apart two metric spaces are as abstract shapes, so that such convergence has a precise meaning. It turns the collection of all compact metric spaces into one big metric space of its own.
It is built in two steps. First, the Hausdorff distance d_H(A, B) between two subsets of a common metric space measures the worst-case mismatch: it is the smallest r such that A lies in the r-neighborhood of B and B lies in the r-neighborhood of A. Second, since two abstract spaces X and Y share no ambient space, you create one: the Gromov-Hausdorff distance d_GH(X, Y) is the infimum, over all isometric embeddings of X and Y into a common metric space Z, of the Hausdorff distance between their images. Equivalently and more usably, d_GH(X, Y) is computed (up to a factor of 2) via correspondences: pair up points of X and Y so that paired distances differ by as little as possible, and minimize that worst distortion. Two spaces are at distance 0 exactly when they are isometric.
Gromov-Hausdorff convergence is the language in which 'a sequence of spaces converges to a limit space' makes sense without any shared embedding, and Gromov's compactness theorem says a family of compact spaces with uniformly bounded diameter and a uniform bound on how many small balls are needed to cover them (uniformly totally bounded) is precompact — every sequence has a convergent subsequence. This is the engine behind taking limits of manifolds: rescaling a group's Cayley graph and letting the scale go to zero gives its asymptotic cone, and collapsing or degenerating Riemannian metrics produce GH limits that may be singular. A caveat: the limit of smooth manifolds need not be a manifold — it can be a lower-dimensional or singular metric space — and curvature bounds (Alexandrov below, or Ricci below) are what control how wild the limit can be.
Consider a sequence of finer and finer regular polygons inscribed in a circle of radius 1, each given its intrinsic (arc-along-the-edges) metric. As the number of sides n -> infinity, these polygons converge in Gromov-Hausdorff distance to the circle with its arc-length metric. No two of them sit in the same space a priori, but the obvious near-isometric correspondences distort distances by an amount going to 0, so d_GH -> 0. The limit is exactly the smooth circle.
Polygons approximating a circle converge to it in the Gromov-Hausdorff metric.
Gromov-Hausdorff limits of manifolds need not be manifolds. Without a lower curvature bound the limit can collapse to a lower dimension or develop singularities, so 'GH-close to a sphere' does not by itself imply 'is a sphere.'