Metric Geometry & Approximation

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Abstractions about metric and pseudometric spaces, geometric means, projections, convergence, packing, embeddings, and approximation problems.

13 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Aleksandrov–Rassias Problem — Ask when a mapping between real normed spaces that preserves one prescribed distance must preserve every distance and therefore be an isometry.
• Ball Tree — Index points in a metric space with a hierarchy of enclosing balls so triangle-inequality lower bounds prune whole subtrees during exact nearest-neighbor and geometric search.
• Delone Set — A metric-space point set with both a positive uniform-separation bound and a finite covering-radius bound, so it is nowhere arbitrarily crowded and nowhere arbitrarily sparse.
• Equilateral Dimension — Measure how large an exactly pairwise-equidistant subset a metric space can support, with the distance scale and attainment convention stated explicitly.
• Fréchet Mean — Any point minimizing expected or empirical squared metric distance to observations, generalizing the Euclidean arithmetic mean to nonlinear metric spaces.
• Gromov–Hausdorff convergence — A convergence notion for metric spaces defined by their Gromov–Hausdorff distance tending to zero after comparison in common ambient metrics.
• Metric Map (Nonexpansive Map) — A function between metric spaces that never increases pairwise distance, equivalently a Lipschitz map with constant at most one.
• Metric projection — Map a point to the set of points in a designated subset that minimize its metric distance, retaining nonexistence and nonuniqueness unless geometry supplies stronger guarantees.
• 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.
• Pseudometric space — Equip a set with a symmetric, nonnegative, triangle-inequality distance that may assign zero separation to distinct points, with metric quotient obtained by identifying zero-distance classes.
• Quasisymmetric map — Control relative metric distortion by requiring every ratio of two distances from a common base point to be bounded through one homeomorphic control function.
• Sphere packing — Arrange nonoverlapping equal-radius balls in a specified ambient space to maximize a declared finite or asymptotic density under explicit boundary, periodicity, and congruence conventions.
• Tarski's Plank Problem — Relate geometric coverage to directional thickness: any finite family of Euclidean planks covering a convex body must have total plank width at least the body's minimum width.