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

**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.**

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

**Are there any other non-trivial regular uniform metric spaces?**@user44191 Provided an affirmative answer to this question. A discrete set of $n$ points such that all distances between them are unique is a regular uniform metric space with degree $n$.**Can we find a regular metric space for every degree $n$?****What can we say about the p-adic numbers $\mathbb{Q}_p$?**They wouldn't be regular in the above sense. Consider for example the set of points that are distance 1 from 0. Then every other number will be a center of these numbers. We can make the same argument for the set of points that are distance $\frac{1}{p}$ from 0. If this was an equivalence relation then from the above to sets we should get that all points are in the same equivalence class. But looking back at the set of points with distance 1 from 0, we see that those points are not the center of it's own set as for example $|5|_3 = 1$ and $|2|_3 = 1$ but $|5-2|_3 \neq 1$. But I guess they are uniform.**What can we say about the surface of $S^n$? What can we say about a torus? A torus of genus 2? etc****Are there any equivalent definitions of uniformity or regularity?****Is there another condition other than regularity that would make this notion an equivalence class?****What can we say about the set of spaces on which this notion is an equivalence class?**

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