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I am currently looking into the lonely runner conjecture

(Consider $k$ runners on a circular track of unit length. At $t = 0$, all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is said to be lonely at time t if they are at a distance of at least $\frac{1}{k}$ from every other runner at time t. The lonely runner conjecture states that each runner is lonely at some time.)

to hopefully learn something new, not to solve it (of course).
I am asking myself if it would be enough to prove, that any runner with speed $v=0$ will always be lonely at some point in time. My thought process is that this would be enough because you can set the speed of every runner to $0$ by subtracting the value of this specific speed from every speed of all the runners, and that should work because all the speed change by some constant.

Is this correct or am I missing something?

SweatyHeater
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  • This type of idea appears to be the content of the observation on that page that in a reformulation of the problem, "the runner to be lonely has zero speed." This type of adjustment cannot set the speed of every runner to $0$ because the speeds are assumed to be pairwise distinct; any given shift will only zero one of the velocities at a time. Cool problem, I had never heard of it. Thanks. – leslie townes Feb 17 '21 at 20:22
  • @saulspatz Nothing abnormal in terms of "relative speeds" that every runner can have. How to use them is another thing. – Jean Marie Feb 17 '21 at 20:36

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