It is well known that the ratio of the circumference of a circle to its diameter in Euclidean geometry is the constant $\pi$. I also understand that in the case of non-euclidean geometry this ratio is in general not a constant.
What I would like to know, is whether or not $d=2r$ holds in these non-euclidean geometries. For the purpose of this question a circle is defined as the set of points with a constant distance (the radius) to a given point, for any metric, and the diameter as the length of the largest distance between any two points on this circle. In particular, I am interested in geometries with a non-constant curvature.
If circles can be defined in a geometry then the radii of a circle will be of the same length, say, $r$. Assume then that in our geometry there are always two points on any circle whose distance, $d$, is maximal. We can say only that $$2r\ge d.$$
As depicted in the figure below, let the diameter of the circle centered at $C$ be the path connecting $A'$ and $B'$. $A'B'$ is the shortest possible path between $A'$ and $B'$ and at the same time it is the longest possible path between any two points that lie on the circle. This path either goes through the center $C$ (then $d=2r$) or it does not. If it does not then $A'C+CB'=2r>d.$ Note that $AB<AC+CB=2r$ if the shortest path connecting $A$ and $B$ does not go through the center.
In the Euclidean, the hyperbolic, and the elliptic geometries the shortest path is straight and $d=2r$. This is because in these geometries the following theorem holds: If we have two triangles $ACB$ and $A'CB'$ and $AC=CB=A'C=CB'$ and the angle $ACB>A'CB'$ then $AB>A'B'$. Also, the straight line determined by $A$ and $B$ is unique.