metricsthickness

Metricsthickness is a term used in theoretical geometry and metric space analysis to describe a coarse measure of how a metric space can be decomposed into line-like pieces. The concept envisions the space as a union of simpler, one-dimensional components and asks how many such components are needed to cover the space without distorting the restricted metrics.

Definition: For a metric space (X,d), the metricsthickness MT(X,d) is the smallest integer k such that X can be partitioned into k subsets X = X1 ∪ ... ∪ Xk, where each Xi is isometric to a subset of the real line with the standard metric. If no finite k suffices, MT(X,d) is defined to be infinite. In short, MT(X,d) measures the minimum number of line-like pieces required to represent the space at the given metric without changing intra-piece distances.

Examples and intuition: If X is already a subset of the real line, MT(X,d) = 1. If X is a union of two disjoint line-like pieces, MT(X,d) ≤ 2. More complex spaces may have higher finite MT values or be infinite, indicating greater departure from one-dimensional structure.

Relations and context: Metricsthickness is related to ideas of dimensionality reduction, path-like decompositions, and notions such as treewidth and ultrametric depth. It provides a coarse alternative to more detailed invariants like the doubling constant or metric dimension, focusing on the number of line-like components needed for a cover.

Computation and use: For finite metric spaces, determining MT can be approached via partitioning algorithms that seek the smallest number of line-like subsets. Exact computation is typically challenging and often impractical for large data sets, so heuristics are common. Potential applications include clustering, network design, and analysis of geo-spatial or road networks.

See also: metric dimension, doubling constant, ultrametric spaces, graph thickness.

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