Convex metric space

In mathematics, convex metric spaces can be thought of, intuitively, as spaces generalizing the idea that given any two points in a space, there exist a third point "between" the two.

Formally, a metric space $(X,d)$ is called metrically convex if for any two distinct points $x$ and $y$ in $X$ there exist a third point $z$ in $X$ distinct from the first two, such that

$d(x,z)+d(z,y)=d(x,y).\,$

Euclidean spaces, that is, the usual three-dimensional space and its analogues for other dimensions, are convex metric spaces. Given any two distinct points $x$ and $y$ in such a space, the set of all points $z$ satisfying the above equality forms the line segment between $x$ and $y,$ which always has other points except $x$ and $y,$ in fact, it has a continuum of points.

Inspired by this example, for any arbitrary metric space $(X,d)$ and any two points $x\neq y$ in $X,$ define the segment $S_{xy}$ between $x$ and $y$ to be the set of all points $z$ in $X$ such that $d(x,z)+d(z,y)=d(x,y).$ Then, $(X,d)$ is a convex metric space if and only if the segment $S_{xy}$ has other elements except $x$ and $y$ for any $x\neq y$ in $X.$

References

• Khamsi, Mohamed A. (2001). An Introduction to Metric Spaces and Fixed Point Theory. Wiley-IEEE. ISBN 0471418250. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Kaplansky, Irving (2001). Set Theory and Metric Spaces. American Mathematical Society. ISBN 0821826948. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)