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.

In a general convex metric space, one can also draw a "continuous segment" between any two points in the space, provided that the space is complete.

To make this precise, a set $S$ in a metric space $(X,d)$ (not necessarily convex for the moment) is called a metric segment between two distinct points $x$ and $y$ in $X,$ if there exists an isometry

$\gamma :[0,1]\to X\,$

such that $\gamma ([0,1])=S,$ $\gamma (0)=x$ and $\gamma (1)=y.$

It is clear that any point in such a metric segment $S$ except for the "endponts" $x$ and $y$ is between $x$ and $y.$ As such, if a metric space $(X,d)$ admits metric segments between any two distinct points in the space, then it is a convex metric space. And the other way around, if $(X,d)$ is a convex metric space, and, in addition, it is complete, then one can connect any $x\neq y$ in $X$ by a segment (which is not necessarily unique).

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.