Diameter and frontier in metric spaces It is well known that in normed vector spaces, ${\rm diam}({\rm Fr}(A)) = {\rm diam}(A)$.
What I'm looking for is a counterexample to this equality in a metric space not deriving from a norm.
With the discrete distance (${\rm d}(x,y)=0$ if $x=y$, else ${\rm d}(x,y)=1$), ${\rm Fr}(A)$ is empty for any $A$, so you could say ${\rm diam}=-\infty$, but it's just a trick. I'm looking for a real counterexample where ${\rm diam}({\rm Fr}(A)) \ne {\rm diam}(A)$.
If you can provide some help, I would be grateful.
\bye
 A: Let $X = S^1 = \{p\in\mathbb{C}: |p|=1\}$, with the standard geodesic metric. Then the subset $A = \{e^{i\vartheta} | \vartheta\in[0, \tfrac{3\pi}2]\}$ has diameter $\pi$, but its boundary has only diameter $\tfrac\pi2$.
  
A: Let
$$X = \{(0, x) : x \in \Bbb{R}\} \cup \{(1, 0)\}$$
and
$$A = \{(0, 0), (1, 0)\}.$$
Note $A \subseteq X \subseteq \Bbb{R}^2$. We consider $X$ as a metric subspace of $\Bbb{R}^2$.
Then the diameter of $A$ is the distance between the two points: $1$. Also, the frontier (or boundary, as I'm used to calling it) of $A$ is $\{(0, 0)\}$, as $(1, 0)$ is isolated in $X$, and hence part of the interior of $A$ (and this is not true for $(0, 0)$). This is a singleton set with diameter $0$. Thus, we have a counterexample.
A: Choose $(\Bbb{R}, d_{\text{std}}) $ and $A=(0, \infty) $.
Then ${\rm Fr}(A)=\partial(A)=\overline{A}\setminus\mathring{A}=\{0\}$
${\rm diam}({\rm Fr}(A)) =0\neq \infty= {\rm diam}(A)$
Your result fails even in normed spaces.
A: This answer is not much different from that of leftaroundabout.
Let $X=\{ (x,y)\in\mathbb{R}^2 \mid x^2+y^2\le 1 \}$ be the unit disk with the usual distance (inherited from $\mathbb{R}^2$). Then $X$ is a compact metric space with diameter $2$.
Consider the compact subset $A=\{ (x,y)\in X \mid x\le 0 \text{ or } y\le 0 \}$. Then the diameter of $A$ is $2$. But the boundary ${\rm Fr}(A)$ of $A$ inside $X$ consists of only two radii, and the diameter of ${\rm Fr}(A)$ is $\sqrt 2$ (realized as the distance between the two points $(1,0)$ and $(0,1)$).

