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Let $X$ be a metric space with the distance function $d$. Given a subset $S \subseteq X$, what are the required conditions on $X$ and $S$ are so that $d(x, \partial S) \leq d(x, \operatorname{ext} S)$ is true for all $x \in S$?

Note that the distances in the above inequality are point-set distances; i.e. $d(x, Y) \equiv \inf_{y \in Y} d(x, y)$. Also, it is easy to see that the given statement is not true in general. For example, if $S$ is a clopen set, it has an empty boundary, which means the distance on the left hand side is infinite.

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  • $\begingroup$ A sufficient condition is that $\operatorname{ext} S = \varnothing \neq \partial S$. But there are other options too. They are not entirely dissimilar, however. $\endgroup$ – Daniel Fischer Feb 16 '14 at 2:50
  • $\begingroup$ @DanielFischer, your answer is of course true but not really what I'm looking for. The intuition that leads to this question is that "you have to cross the boundary before reaching anything outside". I'm thinking there has to be "natural" mathematical property for sets that captures this notion. $\endgroup$ – iheap Feb 16 '14 at 3:05
  • $\begingroup$ D'oh! You have $\leqslant$. I thought it was $<$. $\endgroup$ – Daniel Fischer Feb 16 '14 at 3:09
  • $\begingroup$ It is more about d than S: the property holds if d is a path-metric. $\endgroup$ – Moishe Kohan Feb 16 '14 at 3:26
  • $\begingroup$ @studiosus, will it still hold if $(X, d)$ is locally diffeomorphic to a length space? $\endgroup$ – iheap Feb 16 '14 at 7:50
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This is true for length/path metric spaces and false in general.

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