Continuing from the last post "Lonely Runner Not Lovely",
Runners having the same origin is important. They start at the same place, same time. With that in mind,
Can the next runner added to any point on the unit circle, with arbitrary speed provide a lonely opportunity in the future?
Immediately, this runner cannot land in arc \(\pm \cfrac{1}{n+1}\) around the \(n\) runner, that is,
\(v_{n+1}.t=m_{n+1}\pm\cfrac{1}{n+1}\)
This forbidden region is smaller with large \(n\).
But can this runner, provide an lonely opportunity in the future? The \(n\) runner is the stationary runner at the origin.
Not necessarily. Only the point placement at exactly at \(\cfrac{1}{n+1}\), with making any other assumption.
