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Here, I'm defining $d(x,K) = \inf_{z\in K} d(x,z)$. Since $d$ is continuous and $K$ compact, them there is a point $z\in K$ such that $d(x,K) = d(x,z)$. I know that if $(X,d)$ have the structure of a Length Space with $d$ intrinsic to the length functional, then we can easily show that $z \in \partial K$. My question is: if $(X,d)$ have no such structure, can this be false? Is it possible to have an interior point $z \in K$ that is the minimum?

I'm more interested in the case of a closed ball in $X$ that is compact by hypothesis. Is it possible to construct a compact closed ball in which this happens?

Thank you very much.

Kaitei
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    In general $\partial K$ might be empty (think of the discrete metric) – Arctic Char Sep 20 '21 at 19:52
  • Is the metric necessarily complete? – cjohnson Sep 20 '21 at 19:53
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    Think of a cone un $\mathbb R^3$, with $K$ being its base, a closed disc, and $x$ its summit. Then the distance is achieved only by the the center of the disc, an interior point. – Andrei.B Sep 20 '21 at 19:56
  • @Andrei.B You mean the cone surface union the base correct? I think that answer the question, since it is a (non discrete) compact metric space with a compact ball as a cone base with the euclidean metric. Also shows why the length structure is important in theses cases. Please, write that as an answer so I can accept it :) – Kaitei Sep 20 '21 at 20:18
  • Also thanks for the comments ArcticChar and cjohnson – Kaitei Sep 20 '21 at 20:19

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Start with $X$ obtained from $\mathbb R^2$ by removing the interior of the square $[-1,+1] \times [-1,+1]$: \begin{align*} X &= \{(x,y) \in \mathbb R^2 \mid |x| \ge 1, |y| \ge 1\} \\ K &= [1,2] \times [-1,+1] \\ p &= (-1,0) \end{align*} The point $q = (+1,0) \in K$ is the unique point of $K$ that minimizes the distance to $p$. But $q$ is an interior point of $K$ in the space $X$.

Note that the distance is indeed NOT intrinsic to the length functional: the distance between $p$ and $q$ in $X$ is $2$ but the shortest path in $X$ from $p$ to $q$ has length $4$, going around the periphery of the square $[-1,+1] \times [-1,+1]$.

As for your question regarding a closed ball, do you mean that you want $K$ to be a closed ball? If so then we can modify $K$ easily: take $K$ to be the closed ball in $X$ with center $(1.1,0)$ and radius $.2$. Instead of a rectangular shape as in the $K$ above, this $K$ would be the closed Euclidean disc with that same center and radius, with a portion cut off that is bounded by the chord where that disc intersects the line $x=1$. All interior points of that chord, including $(+1,0)$, will be interior points of $K$.

Lee Mosher
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