Another Theorem By Changing The Question

 Suppose it is established that $n$ runner running about a unit circle with arbitrary constant speeds, $v_i$, one just experienced a moment of loneliness, and is at least $\cfrac{1}{n}$ away from the closest runner.

We add the next $(n+1)$ runner, with relative speed zero, where all other runners take reference, right in between and achieve a least distance of $\cfrac{1}{2n}$.  This $(n+1)$ runner is at the origin zero (starting point).  The added $(n+1)$ runner still has arbitrary speed, distinct and constant.

This is the trivial case of $n=3$.

Since     $\cfrac{1}{2n}\lt\cfrac{1}{(n+1)}$

for $n\gt 2$

Adding the next $(n+1)$ runner becomes trivial, and with the trivial case of $n=3$, by induction the conjecture is true.  The constraint is still a least distance of $\cfrac{1}{n}$, it has not been changed to $\cfrac{1}{2n}$.

What?  Using relative speeds is not suppose to change any aspect of the analogy.

 Happy? 

Note: All speeds changed by a constant, but the transition of time has no bearing here. The origin is rotated by a constant.  The circle is rotating at the reference speed, speed of the $(n+1)$ runner.  Starting time is later for all.