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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 $RC$?

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