A quadratic lower bound for the convergence rate in the one-dimensional Hegselmann-Krause bounded confidence dynamics

Let f_{k}(n) be the maximum number of time steps taken to reach equilibrium by a system of n agents obeying the k-dimensional Hegselmann-Krause bounded confidence dynamics. Previously, it was known that Ω(n) = f_{1}(n) = O(n^3). Here we show that f_{1}(n) = Ω(n^2), which matches the best-known lower bound in all dimensions k >= 2.