I'm reading a book called Geometry of quantum states. On page 18 the authors introduce the usual notion of metric on a vector space and continue to discuss Minkowski distance.
A metric $d$ on a vector space $V$ is called a Minkowski distance if $$ \lambda d(x,y) = d(\lambda x, \lambda y) $$ for all positive reals $\lambda$.
According to the Authors, an easy consequence of the definition is that for any $x, y \in V$ and any convex combination $z$ of $x$ and $y$, we have $$ d(x,y) = d(x,z) + d(z, y). $$ I spent some time trying to prove this. This is indeed trivial if the metric is induced by a norm, but I failed to prove this in general.
Now I suspect that either I have overlooked something in the definition, or the author's have slipped here, since the metric $d(x,y) = |x| + |y|$ for $x \neq y$ and $d(x,y) = 0$ for $x = y$ on $\mathbb{R}$ seems to be a Minkowski distance, but it doesn't seem to satisfy the second equation above.
Is anyone familiar with this topic?
