http://www.cs.toronto.edu/~avner/teaching/S6-2414/LN1.pdf
I was reading the link above and am stuck on Example 2.2. Why should f(d), f(a) and f(b) be collinear in $R^k$ ? I understand that the triangle inequality has to hold, but why can't the structure remain as it is?
The notes you cite say that "the $\triangle$-inequality is tight". As there is no indication of drunkenness in the accompanying figure, I think what is meant is that equality holds in the triangle inequality for the points in question: i.e., you have $d(x, z) = d(x, y) + d(y, z)$ not just $d(x, z) \le d(x, y) + d(y, z)$. This can only happen in the euclidean space $\Bbb{R}^k$ if $x$, $y$ and $z$ are collinear.
