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Inequalities on Six Points in a $\mathrm{CAT}(0)$ Space

Tetsu Toyoda

TL;DR

The article addresses the problem of characterizing when a finite metric space embeds into a CAT$(0)$ space through quadratic metric inequalities. It introduces and proves an explicit 6-point CAT$(0)$ quadratic inequality parameterized by $a,b,c,s,t\in(0,1)$ with $a\le s$, showing this inequality holds in all CAT$(0)$ spaces and cannot be deduced from $5$-point or $4$-point CAT$(0)$ conditions. By leveraging Lebedeva’s $6$-point constructions and barycenter/variance techniques, it demonstrates that there exist $6$-point spaces for which all $5$-point subsets embed into CAT$(0)$ spaces and yet the 6-point inequality fails for certain parameter choices, highlighting a separation in the hierarchy of inequalities. The work also connects to ANN inequalities and graph-comparison notions such as the $O_3$-comparison, providing corollaries and propositions that clarify the landscape of 6-point embeddability and its relation to known sufficient conditions.

Abstract

We establish a family of inequalities that hold true on any $6$ points in any $\mathrm{CAT}(0)$ space. We prove that the validity of these inequalities does not follow from any properties of $5$-point subsets of $\mathrm{CAT}(0)$ spaces. In particular, the validity of these inequalities does not follow from the $\mathrm{CAT}(0)$ $4$-point condition.

Inequalities on Six Points in a $\mathrm{CAT}(0)$ Space

TL;DR

The article addresses the problem of characterizing when a finite metric space embeds into a CAT space through quadratic metric inequalities. It introduces and proves an explicit 6-point CAT quadratic inequality parameterized by with , showing this inequality holds in all CAT spaces and cannot be deduced from -point or -point CAT conditions. By leveraging Lebedeva’s -point constructions and barycenter/variance techniques, it demonstrates that there exist -point spaces for which all -point subsets embed into CAT spaces and yet the 6-point inequality fails for certain parameter choices, highlighting a separation in the hierarchy of inequalities. The work also connects to ANN inequalities and graph-comparison notions such as the -comparison, providing corollaries and propositions that clarify the landscape of 6-point embeddability and its relation to known sufficient conditions.

Abstract

We establish a family of inequalities that hold true on any points in any space. We prove that the validity of these inequalities does not follow from any properties of -point subsets of spaces. In particular, the validity of these inequalities does not follow from the -point condition.

Paper Structure

This paper contains 7 sections, 8 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Andoni-Naor-Neiman inequalities
  3. Graph comparison
  4. Towards a characterization of $6$-point subsets of $\mathrm{CAT}(0)$ spaces
  5. Organization of the paper
  6. Preliminaries
  7. Proof of Theorem \ref{['6-pt-ineq-th']}

Key Result

Theorem 1.2

A metric space $X$ with $|X|\leq 5$ admits a distance-preserving embedding into a $\mathrm{CAT}(0)$ space if and only if $X$ satisfies the $\boxtimes$-inequalities.

Figures (3)

  • Figure 1: The octahedron graph $O_3$.
  • Figure 2: Lebedeva's six points in $\mathbb{R}^3$.
  • Figure 3: The points $x_0 ,x_1 ,y_0 ,y_1 ,z_0 ,z_1$ in $\mathbb{R}^3$.

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 1.2: toyoda-five
  • Theorem 1.3
  • Theorem 1.4: Andoni, Naor and Neiman ANN
  • Corollary 1.5
  • Definition 1.6: Gromov Gr2
  • Definition 1.7: Lebedeva, Petrunin and Zolotov LPZ
  • Proposition 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 7 more