# Is there a name for this notion of “radius of compactness” in a metric space?

I was proving some result about Riemannian manifolds that led me to introduce the following definition:

Let $$M$$ be a metric space and $$x \in M$$. Define the "radius of compactness" $$RC(x)$$ to be the $$\sup$$ of $$R \geq 0$$ such that the closed ball $$\bar B(x, R)$$ is compact.

I like to think of it as a measure of how far a point is from the "boundary" of $$M$$.

It satisfies nice properties such as:

• $$|RC(y) - RC(x)| \leq d(x,y)$$
• If $$M$$ is locally compact and $$K \subset M$$ compact, then there exists $$\epsilon > 0$$ such that $$RC(x) \geq \epsilon$$ for all $$x \in K$$.

Is there a standard name for this $$RC$$?