Assume that we have a metric space $(S,d)$ and points $a,b,c \in S$ which statisfy the following conditions:
- for all $x \in S$, $d(a,x) \leq d(a,b)$,
- for all $y\in S$, $d(b,y) \leq d(b,c)$.
Does the following statement follow from these two assumptions?
- for all $x,y\in S$, $d(x,y) \leq d(b,c)$
In words: if $b$ is a furthest point from $a$ and $c$ is a furthest point from $b$, is $d(b,c)$ the maximum possible distance between points in $S$?
In case someone is interested in the motivation: I was trying to see how much graph structure is needed in the proof that using two BFSes we can find the diameter of a finite connected graph:
- Start from $a$ and find a vertex $b$ with maximum distance from $a$.
- Start from $b$ and find a vertex $c$ with maximum distance from $b$.
- $d(b,c)$ is the diameter of the graph.
The algorithm doesn't work correctly. Here is a counterexample: