In why-the-triangle-inequality
I found the statement:
for example if $d(a,b)$ measures the "length" of the "shortest path" between points $a$ and $b$ (and this can be interpreted quite abstractly, for example if $a$ and $b$ are states of some physical system and $d(a,b)$ describes the amount of energy that needs to be expended to get from $a$ to $b$), then $d(a,c)≤d(a,b)+d(b,c)$ because an optimal "path" from $a$ to $b$, together with an optimal "path" from $b$ to $c$, can be no better than the optimal "path" from $a$ to $c$.
Until now I assumed that minimal paths do not fullfill the triangle inequality. Does someone know, how to prove this or where to find it proved? I'm working on an thesis in informatics (graph theory) and need evidence for this statement, e.g. for minimal paths in weighted Delaunay-Graphs.
Definitions:
Maybe this is also relevant: I use Dijkstra's Shortest-Path Algorithm to calculate g(l,j)