In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and

Felix Hausdorff Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...

, is a notion for convergence of

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

s which is a generalization of Hausdorff convergence.

Gromov–Hausdorff distance

The Gromov–Hausdorff distance was introduced by David Edwards in 1975, and it was later rediscovered and generalized by Mikhail Gromov in 1981. This distance measures how far two

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

metric spaces are from being isometric. If ''X'' and ''Y'' are two compact metric spaces, then ''d_GH'' (''X'', ''Y'') is defined to be the

infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...

of all numbers ''d''_''H''(''f''(''X''), ''g''(''Y'')) for all metric spaces ''M'' and all isometric embeddings ''f'' : ''X'' → ''M'' and ''g'' : ''Y'' → ''M''. Here ''d''_''H'' denotes

Hausdorff distance In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a me ...

between subsets in ''M'' and the ''isometric embedding'' is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no compact Riemannian manifold admits such an embedding into

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

of the same dimension. The Gromov–Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, called Gromov–Hausdorff space, and it therefore defines a notion of convergence for

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

s of compact metric spaces, called Gromov–Hausdorff convergence. A metric space to which such a sequence converges is called the Gromov–Hausdorff limit of the sequence.

Some properties of Gromov–Hausdorff space

The Gromov–Hausdorff space is path-connected, complete, and separable. It is also geodesic, i.e., any two of its points are the endpoints of a minimizing geodesic. In the global sense, the Gromov–Hausdorff space is totally heterogeneous, i.e., its isometry group is trivial, but locally there are many nontrivial isometries.

Pointed Gromov–Hausdorff convergence

The pointed Gromov–Hausdorff convergence is an analog of Gromov–Hausdorff convergence appropriate for non-compact spaces. A pointed metric space is a pair (''X'',''p'') consisting of a metric space ''X'' and point ''p'' in ''X''. A sequence (''X_n, p_n'') of pointed metric spaces converges to a pointed metric space (''Y'', ''p'') if, for each ''R'' > 0, the sequence of closed ''R''-balls around ''p_n'' in ''X_n'' converges to the closed ''R''-ball around ''p'' in ''Y'' in the usual Gromov–Hausdorff sense.

Applications

The notion of Gromov–Hausdorff convergence was used by Gromov to prove that any

discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...

with polynomial growth is virtually nilpotent (i.e. it contains a nilpotent subgroup of finite index). See

Gromov's theorem on groups of polynomial growth In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of ''polynomial'' growth, as those groups which have nilpotent subgroups of finite index. Statement ...

. (Also see D. Edwards for an earlier work.) The key ingredient in the proof was the observation that for the

Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...

of a group with polynomial growth a sequence of rescalings converges in the pointed Gromov–Hausdorff sense. Another simple and very useful result in

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

is Gromov's compactness theorem, which states that the set of Riemannian manifolds with

Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...

≥ ''c'' and

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...

≤ ''D'' is

relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sin ...

in the Gromov–Hausdorff metric. The limit spaces are metric spaces. Additional properties on the length spaces have been proven by Cheeger and Colding. The Gromov–Hausdorff distance metric has been applied in the field of computer graphics and computational geometry to find correspondences between different shapes. It also has been applied in the problem of

motion planning Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination. The term is use ...

in robotics. The Gromov–Hausdorff distance has been used by Sormani to prove the stability of the Friedmann model in Cosmology. This model of cosmology is not stable with respect to smooth variations of the metric. In a special case, the concept of Gromov–Hausdorff limits is closely related to large-deviations theory. The Gromov–Hausdorff distance metric has been used in neuroscience to compare brain networks.

References

* M. Gromov. ''Metric structures for Riemannian and non-Riemannian spaces'', Birkhäuser (1999). (translation with additional content). {{DEFAULTSORT:Gromov-Hausdorff convergence Metric geometry Riemannian geometry Convergence (mathematics)