Given the metric $d_p$ on the real plane, i.e.

$$ d_p(x,y) = d_p((x_1, y_1), (x_2, y_2)) = [|x_1 - x_2|^p+ |y_1 - y_2|^p]^{1/p} $$

for which values of $p$ ($\geq 1$) is it true that the following set is the usual line segment between points $x$ and $y$ :

$$ \{ z \in \mathbb{R}^2 \;|\; d_p(x,z) + d_p(z,y) = d_p(x,y)\}$$

more generally is there a simple meaningful characterization for a general metric $d$ for which the above is the usual line segment ?