What does a segment in the plane as a metric space defined by a $p$-norm look like?

Question

In the metric space $\mathbb R^2$ with the metric $d$ defined by $d(x,y)= (|x_1-y_1|^p+|x_2-y_2|^p)^{1/p}$, where $p\gt1$ is a real number, like what does the set of all $m\in \mathbb R^2$ with $d(a,m)+d(m,b)=d(a,b)$ look, where $a$ and $b$ are two arbitrary points of $\mathbb R^2$? I think for all $p$’s it is a straight line segment as in the obvious case $p=2$ but I do not know how to deal with the other values of $p$.