 Solution

The stated $d$ is a metric; the triangle inequality follows by routing through $x_0$, with equality in the only nontrivial case. Star-shaped sets are path-connected by joining each point to $x_0$. Along each radial segment, the fundamental theorem and $\Vert Df\Vert \le M$ give $\Vert f(x)-f(x_0)\Vert \le M\Vert x-x_0\Vert$; adding the two bounds proves the $Md(x,y)$ estimate. If $Df=0$, both radial differences vanish, so $f$ is constant.