Gromov-Hausdorff convergence

Hausdorff convergence is a term from mathematics, is described by the that compact subsets of (or of a general metric space ) approach a limit amount. It is used in the fractal geometry for the construction of fractals and in differential geometry for guiding contrary evidence.

Held generally, the notion of Gromov - Hausdorff convergence, where convergence of arbitrary sequences of compact metric spaces (not necessary subsets of a given space ) describes.

Definition

Be a metric space and a sequence of compact subsets. The sequence converges to the compact set if

Applies. Herein, the Hausdorff distance.

Advertised does this definition: converges if there is one for all, such that for all: is the environment of and is located in the environment of.

Properties

Limits of sequences of convex sets in Euclidean space are convex, limits of sequences of related quantities are contiguous. In contrast, the limit of a sequence path-connected spaces need not always be path-connected.

The result right in the picture is a sequence of tori, which converges to a circle. The limit of a sequence homeomorphic spaces must not necessarily homeomorphic to the individual sequence elements be, he may even have lower dimension.

The result right in the picture is a series of curves of length which converges to a curve of length. The length of the curves is thus not continuous with respect to Hausdorff convergence, but is above steady. Higher-Dimensional volumes of surfaces, bodies etc. are generally neither below nor above continuous with respect to Hausdorff convergence.

Compactness theorem

By a theorem of Blaschke following compactness criterion applies to the Hausdorff convergence.

Be desired, a ball of radius, and a sequence of compact sets, then there is a Hausdorff convergent subsequence.