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A generalized metric-type structure with some applications

Hallowed O. Olaoluwa, Aminat O. Ige, Johnson O. Olaleru

TL;DR

The paper introduces O-metric spaces as a unifying generalization of metric-type spaces by replacing the triangle inequality with a binary operation $\textnormal{o}$ on distances, and distinguishes upward ($I_a\subset[a,\infty)$) and downward ($I_a\subset[0,a]$) variants. It develops the topology induced by O-metrics, establishes when convergence, Cauchy sequences, and openness behave analogously to classical metric spaces, and bridges to familiar spaces via products and transformations. A Banach-type fixed point theorem is proved for contractive-like maps in O-metric spaces, yielding classical results as corollaries and enabling broader applicability. The paper then develops sharp polygon inequalities and a suite of applications, including refined triangle inequalities, interpretations in terms of $s$-relaxed and $s$-constrained triangle inequalities for b-metrics and infinite symmetric matrices, and concrete constructions that connect distance inequalities to matrix analyses.

Abstract

The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss convergence, sequential continuity, first countability and T$_2$ separation. The topology of an O-metric space can be generated by an upward O-metric on the space hence the focus on upward O-metric spaces. A theorem on the existence and uniqueness of a fixed point of some contractive-like map is proved and related with some other well known fixed point results in literature. Applications to the estimation of distances, polygon inequalities, and optimization of entries in infinite symmetric matrices are also given.

A generalized metric-type structure with some applications

TL;DR

The paper introduces O-metric spaces as a unifying generalization of metric-type spaces by replacing the triangle inequality with a binary operation on distances, and distinguishes upward () and downward () variants. It develops the topology induced by O-metrics, establishes when convergence, Cauchy sequences, and openness behave analogously to classical metric spaces, and bridges to familiar spaces via products and transformations. A Banach-type fixed point theorem is proved for contractive-like maps in O-metric spaces, yielding classical results as corollaries and enabling broader applicability. The paper then develops sharp polygon inequalities and a suite of applications, including refined triangle inequalities, interpretations in terms of -relaxed and -constrained triangle inequalities for b-metrics and infinite symmetric matrices, and concrete constructions that connect distance inequalities to matrix analyses.

Abstract

The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss convergence, sequential continuity, first countability and T separation. The topology of an O-metric space can be generated by an upward O-metric on the space hence the focus on upward O-metric spaces. A theorem on the existence and uniqueness of a fixed point of some contractive-like map is proved and related with some other well known fixed point results in literature. Applications to the estimation of distances, polygon inequalities, and optimization of entries in infinite symmetric matrices are also given.
Paper Structure (16 sections, 26 theorems, 35 equations, 4 figures)

This paper contains 16 sections, 26 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. O-metric spaces: topological structure
  3. Relating types of O-metric spaces
  4. Generating new O-metrics from existing ones
  5. Upward O-metric spaces and metric spaces
  6. Upward O-metric spaces and downward O-metric spaces
  7. Finite product of O-metrics
  8. Topology induced by an O-metric
  9. Convergence and sequential continuity
  10. Openness of balls and Hausdorff property
  11. Generating the O-metric topology and relating properties $(C_i)$, $(E_i)$, $(U_i)$, $i=1,2$
  12. A fixed point theorem on O-metric spaces
  13. Polygon inequalities and Applications
  14. A more precise triangle inequality
  15. $s$-relaxed triangle inequalities and triangle $\textnormal{o}$-inequalities in some b-metrics
  16. ...and 1 more sections

Key Result

Proposition 2.9

Let $X$ be a non-empty set. Consider $a,b \in \mathbb{R}_+$ contained respectively in intervals $I_a$ and $J_b$ of non-negative real numbers. Let $\theta:\mathbb{R}_+ \to \mathbb{R}_+$ be a non-decreasing function, bijective on $I_a$, with $\theta(a)=b$ and $\theta(I_a)=J_b$, and let $\textnormal{o} Then given functions $d_{\textnormal{o}_a}: X \times X \to I_a$ and $d_{\textnormal{o}_b}: X \times

Figures (4)

  • Figure 1: Illustration of condition (\ref{['complicated1']}).
  • Figure 2: Circles of diameter the segment $x_1x_3$ (red), $x_1x_2$ (green) and $x_2x_3$ (blue). The area of the circle in red is not greater than 2 times the sum of the areas of the 2 other circles (left figure), with equality when $x_1,x_2,x_3$ are collinear and $d(x_1,x_2)=d(x_2,x_3)=\sqrt{2}$ (right figure).
  • Figure 3: $z=2(x+y)-(\sqrt{x}+\sqrt{y})^2$.
  • Figure 4: The terms of the sequence $\{x_n\}$ in the case $r=\frac{1}{2}$, are the endpoints of the segments, starting from the least, clockwise.

Theorems & Definitions (74)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • ...and 64 more