Suppose $(X,d)$ is a metric space with diameter $\sup \{ d(x,y) \colon x,y \in X\}=1$.
Call the point $x \in X$ an edge point to mean that $d(x,y)=1$ for some $y \in X$.
Call the metric space round to mean that every point is an edge point.
Obviously the connected examples that spring to mind are circles and spheres of unit diameter. I wonder are there any other examples?
I tried to construct one in the plane to see what must be done and what can go wrong. So start with two points a unit distance apart. Then everything is within a distance $1$ of $x$ and everything is with a distance $1$ of $y$. That means the space must be contained in the "eye shape."
But I cannot see any way to restrict further, other than drawing another "eye" between other pairs of points . Only I don't know where those points are allowed to lie in the first place.
The yellow circle is a possible round space containing both points. Are there any others?
Do spaces of this type have another name?
Examples .....
(1) A discrete space with the discrete metric: $d(x,y)=1$ whenever $x\ne y.$
(2) Let $ABC$ be an equilateral triangle of side length $1.$ Draw a circular arc centered at $A,$ joining $B$ to $C.$ And likewise draw an arc centered at $B,$ from $C$ to $A,$ and an arc centered at $C,$ from $A$ to $B.$ The union of the $3$ arcs is an example.