Metric Geometry¶
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13 domain-specific abstractions whose origin domain is Metric Geometry.
- CAT(k) space — A geodesic metric space whose triangles are no thicker than comparison triangles in the constant-curvature model space of curvature k, within the prescribed perimeter range.
- Covering number — The minimum number of radius-r balls required to cover a specified subset of a metric or pseudometric space.
- Doubling space — A metric space in which every radius-r ball can be covered by at most a fixed number of radius-r/2 balls, giving a scale-uniform finite doubling dimension.
- Equivalence of metrics — A relation between metrics that captures equality of induced topology, uniformity, Lipschitz structure, or another declared level of geometric behavior.
- Gromov's compactness theorem (geometry) — A precompactness theorem for families of compact metric spaces with uniform diameter and covering-number bounds under Gromov–Hausdorff convergence.
- Injective metric space — A metric space into which every nonexpansive map from a subspace extends nonexpansively over the containing space, equivalently a hyperconvex space.
- Isometry group — The group of all bijective self-maps of a metric space that preserve every distance, with composition encoding the space's exact metric symmetries.
- Metric Map (Nonexpansive Map) — A function between metric spaces that never increases pairwise distance, equivalently a Lipschitz map with constant at most one.
- Metric Space Aimed at Its Subspace — A metric superspace whose distance differences to points of a distinguished subspace approximate every ambient pair distance arbitrarily closely.
- Polyhedral space — A metric space assembled by gluing constant-curvature simplices isometrically along compatible faces.
- Triangle inequality — The distance or norm axiom stating that a direct separation is no greater than the length of any two-step path, d(x,z)≤d(x,y)+d(y,z).
- Ultrametric space — A metric space satisfying the strong triangle inequality, so every triangle is isosceles with its two largest distances equal and balls form a nested hierarchy.
- Uniformly disconnected space — A metric space with one scale-independent constant preventing any two distinct points from being joined by a chain of sufficiently small relative steps.