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I have been away from math for a while so be gentle if this is not very rigorous or if I am redefining already defined objects. The essential question that started me on this was the following. In the plane every circle has one center. On the surface of the sphere every circle has two centers. Okay, so let's formalize this notion.

Let $(X, d)$ be a metric space and let $C_r(x) = \{y \in X \ | \ d(x-y) = r \}$. Let $D \subseteq \mathbb{R}$ be the set of all distances between any two points of $X$. Then consider the following function:

$$f: X \times D \rightarrow P(X)$$

$$(x, r) \mapsto \{ z \in X \ | \ \exists s \in D \text{ with } C_s(z) = C_r(x) \}$$

We know that $x \in f(x,r), \forall (x,r) \in X \times D$. So this set is never empty and contains the original point. It is also reflexive in the sense that if $y \in f(x,r)$ for some $r \in D$ then $x \in f(y,s)$ for some $s \in D$. It would be nice if we could make this an equivalence relation. We can say that a metric space $(X,d)$ is regular if $f(x,r) = f(x,s) \ \forall r,s \in \mathbb{D}$. For a regular space $(X,d)$ we can define the following equivalence relation:

$$x \sim y \iff \exists r,s \in D \text{ with } C_r(x) = C_s(y)$$

Returning to our examples:

a) $\mathbb{R}^n$ with the metric induced by any $\mathbb{L}_p$ norm. Every equivalence class contains only the point itself.

b) The surface of a sphere with the "shortest path on the surface" distance. The equivalence classes are the sets of antipodal points.

c) A cone. I believe the center of the base and the top of the cone are in the same equivalence class, while every other point is in its own class.

We can say that a regular metric space is uniform if every equivalence class is of equal size. For a regular uniform metric space we can call the size of the equivalence class the degree of it. So that the first two examples would be regular uniform metric spaces but the 3rd would be regular but not uniform. The degree of $\mathbb{R}^n$ would be 1, while the degree of the surface of the unit sphere would be 2.

So here are some obvious questions I haven't been able to find answers to.