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, and f ...
, 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 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 metric ...
.
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 Britis ...
metric spaces are from being
isometric
The term ''isometric'' comes from the Greek for "having equal measurement".
isometric may mean:
* Cubic crystal system, also called isometric crystal system
* Isometre, a rhythmic technique in music.
* "Isometric (Intro)", a song by Madeon from ...
. If ''X'' and ''Y'' are two compact metric spaces, then ''d
GH'' (''X'', ''Y'') is defined to be the
infimum 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 met ...
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 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
sequences 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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
, 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 with
polynomial growth is virtually nilpotent (i.e. it contains a
nilpotent subgroup of finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
). See
Gromov's theorem on groups of polynomial growth. (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 Cayl ...
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 is
Gromov's compactness theorem, which states that
the set of Riemannian manifolds with
Ricci curvature ≥ ''c'' and
diameter ≤ ''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 (sinc ...
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 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)