Game extensions of floppy graph metrics

Abstract

A $graph$ $metric$ on a set $X$ is any function $d:E_d\to {\mathbb{R}}_{+}:=\{x\in \mathbb{R}:x>0\}$ defined on a connected graph $E_d\subseteq [X{]}^2:=\{A\subseteq X:|A|=2\}$ and such that for every $\{x,y\}\in E_d$ we have $d(\{x,y\})\le \hat{d}(x,y):=\inf \{{\sum }_{i=1}^{n}d(\{x_{i-1},x_i\}):\{x,y\}=\{x_0,x_n\}\wedge \{\{x_{i-1},x_i\}:0<i\le n\}\subseteq E_d\}$. A graph metric $d$ is called a $full$ $metric$ on $X$ if $E_d=[X{]}^2$. A graph metric $d:E_d\to {\bar{\mathbb{R}}}_{+}$ is $floppy$ if $\hat{d}(x,y)>\check{d}(x,y:=\sup \{d(\{a,b\})-\hat{d}(a,u)-\hat{d}(b,y):\{a,b\}\in E_d\}$ for every $x,y\in X$ with $\{x,y\}\notin E_d$. We prove that for every floppy graph metric $d:E_d\to {\mathbb{R}}_{+}$ on a set $X$, every points $x,y\in X$ with $\{x,y\}\notin E_d$, and every real number $r$ with $\frac{1}{3}\check{d}(x,y)+\frac{2}{3}\hat{d}(x,y)\le r<\hat{d}(x,y)$ the function $d\cup \{⟨\{x,y\},r⟩\}$ is a floppy graph metric. This implies that for every floppy graph metric $d:E_d\to {\mathbb{R}}_{+}$ with countable set $[X{]}^2\setminus E_d$ and for every indexed family $(F_e{)}_{e\in [X{]}^2\setminus E_d}$ of dense subsets of ${\mathbb{R}}_{+}$, there exists an injective function $r\in {\prod }_{e\in [X{]}^2\setminus E_d}{F}_e$ such that $d\cup r$ is a full metric. Also, we prove that the latter result does not extend to partial metrics defined on uncountable sets.

18 pages

References in corpus (2)

• On the uniqueness of continuation of a partially defined metric
• Banakh spaces and their geometry