2026/07/17 by Hyunwoo Lee
#math.CO #math.DS #math.MG
Motivated by the celebrated Lonely Runner Conjecture, we study a variant in which the runners have time-dependent velocities. Let n ≥ 3 runners start from the same point on the unit circle, where each runner i∈[n] has a locally integrable velocity function νi∈ L1loc(ℝ>0). Assume that their velocities are strictly ordered almost everywhere and that the relative distance between every pair diverges. We prove that each of the slowest and fastest runners is at a distance strictly larger than 2-n+1 from every other runner at some time. Moreover, we show that the distance 2-n+1 is optimal. On the other hand, we construct examples in which every intermediate runner remains arbitrarily close to another runner at all times. As a consequence, we also obtain a sharp nonlinear analogue of a classical theorem of Schoenberg on billiard ball motion in the unit cube.