Floppy graph metrics with countably many missing edges can be extended to full metrics by choosing lengths from dense subsets of positive reals in the interval between one-third check d plus two-thirds hat d and hat d.
Banakh spaces and their geometry
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abstract
Following Will Brian, we define a metric space $X$ to be $Banakh$ if all nonempty spheres of positive radius $r$ in $X$ have cardinality $2$ and diameter $2r$. Standard examples of Banakh spaces are subgroups of the real line. In this paper we study the geometry of Banakh spaces, characterize Banakh spaces which are isometric to subgroups of the real line, and also construct Banakh spaces $(X,d)$ which do not embed into the real line and have a prescribed distance set $d[X^2]$.
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math.CO 1years
2023 1verdicts
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Game extensions of floppy graph metrics
Floppy graph metrics with countably many missing edges can be extended to full metrics by choosing lengths from dense subsets of positive reals in the interval between one-third check d plus two-thirds hat d and hat d.