Let $(X,d)$ be a metric space, and define $\rho$ to be the largest distance of any $x\in X$ to its nearest neighbor.

Formally, $$ \rho = \sup_{x \in X}~ d(x, X \setminus \{x\}). $$

Does this quantity have a name? It's zero in continuous spaces and is only interesting in discrete ones.

  • $\begingroup$ If I name it, I might call it “largest isolation.” $\endgroup$ – Tsuyoshi Ito Apr 26 '11 at 17:29
  • $\begingroup$ I was thinking "isolation distance". $\endgroup$ – Aryeh Apr 26 '11 at 17:34

Sloan calls this the "covering radius" when distributing points on a sphere. http://www2.research.att.com/~njas/coverings/

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  • $\begingroup$ Covering radius is slightly different because it is defined in terms of a metric space X and its (usually finite) subset S, whereas in this question there is only one metric space. $\endgroup$ – Tsuyoshi Ito Apr 26 '11 at 18:11

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