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

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

that is, the triangle inequality becomes an equality. 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.

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 there exist an infinite number of points between $x$ and $y$. Moreover, if $(X,d)$ is a complete metric space, one can join $x$ to $y$ by a "segment" which is made up of a continuum points between $x$ and $y.$

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.