Convex metric space

In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints.

Formally, consider a metric space $(X,d).$ Let $x$ and $y$ be two points in $X.$ A point $z$ in $X$ is said to be between $x$ and $y$ if all three points are distinct, and

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

A convex metric space is a metric space $(X,d)$ such that for any two points $x\neq y$ in $X$ there exists $z$ in $X$ lying between $x$ and $y.$

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 $X.$

Let $(X,d)$ is a convex metric space and consider two points $x\neq y$ in $X.$ Let $z$ be a point between $x$ and $y.$ Then, there will exist a point $w$ between $x$ and $z.$ One can then check using the triangle inequality that $w$ is also between $x$ and $y.$ By repeatedly applying this argument it follows that the segment $S_{xy}$ has in fact an infinite number of points. In fact, if $(X,d)$ is complete, one can prove that $S_{xy}$ has a continuum of points, just like in Euclidean spaces.

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)