[Paper Review] Gromov-Hausdorff Distance Between Segment and Circle

Yibo Ji, Alexey Avgustinovich Tuzhilin|arXiv (Cornell University)|Jan 14, 2021

Geometric and Algebraic Topology1 references4 citations

TL;DR

This paper computes the exact Gromov–Hausdorff distance between a line segment of length λ and the unit circle in the Euclidean plane. The authors introduce novel concepts such as round metric spaces and nonlinearity degree, and derive a piecewise formula for the distance: π/2 − λ/4 for λ ≤ 2π/3, π/3 for 2π/3 ≤ λ ≤ 5π/3, and (λ − π)/2 for λ ≥ 5π/3, resolving a long-standing open problem in metric geometry.

ABSTRACT

We calculate the Gromov--Hausdorff distance between a line segment and a circle in the Euclidean plane. To do that, we introduced a few new notions like round spaces and nonlinearity degree of a metric space.

Motivation & Objective

• To determine the exact value of the Gromov–Hausdorff distance between a line segment and a circle in the Euclidean plane.
• To address the longstanding challenge of computing explicit GH-distances for simple yet non-trivial metric spaces.
• To introduce and utilize new geometric invariants such as round spaces and nonlinearity degree to analyze metric distortions.
• To establish a complete piecewise formula for the GH-distance as a function of segment length λ.
• To demonstrate the applicability of correspondence-based methods and geometric covering arguments in metric comparison.

Proposed method

• The authors define a correspondence R between the segment Iλ and the circle S¹, parameterized by angular and linear coordinates.
• They analyze the distortion of the correspondence using geometric covering arguments, particularly focusing on how segments in the parameter space map to regions in the product space.
• The proof employs a geometric construction involving rotated rectangles and cross-shaped regions to cover the correspondence, ensuring distortion bounds.
• Key inequalities involving distances such as |AC|, |CC′|, and |BB′| are used to verify coverage and distortion control.
• The method relies on Theorem 5.1, which links distortion bounds to the existence of certain separated subsets, enabling the derivation of lower bounds.
• The authors use the equivalence between the GH-distance and the infimum of distortion over closed correspondences to establish tight bounds.

Experimental results

Research questions

• RQ1 What is the exact Gromov–Hausdorff distance between a line segment of length λ and the unit circle in the Euclidean plane?
• RQ2 How can the nonlinearity and geometric structure of the circle be leveraged to bound the distortion of correspondences with a segment?
• RQ3 Can a piecewise formula for the GH-distance be derived based on the length λ of the segment?
• RQ4 What role do geometric covering arguments and metric invariants like the nonlinearity degree play in computing GH-distances?
• RQ5 How do the intrinsic metric and topological properties of the circle and segment affect the distortion of optimal correspondences?

Key findings

• The Gromov–Hausdorff distance between a segment of length λ and the unit circle is π/2 − λ/4 for 0 ≤ λ ≤ 2π/3.
• For 2π/3 ≤ λ ≤ 5π/3, the distance stabilizes at π/3, indicating a plateau in metric dissimilarity.
• For λ ≥ 5π/3, the distance grows linearly as (λ − π)/2, reflecting increasing segment length.
• The paper establishes that the minimal distortion correspondence is achieved via a carefully constructed piecewise linear and angular mapping.
• The nonlinearity degree of the circle is shown to be a critical factor in preventing lower distortion, leading to the π/3 lower bound.
• The result confirms that the GH-distance is continuous and piecewise-smooth in λ, with distinct phases corresponding to different geometric regimes.