Let F_n be the empirical distribution function (df) pertaining to independent random variables with continuous df F. We investigate the minimizing point (formula present) of the empirical process F_n − F₀, where F₀ is another df which differs from F. If F and F₀ are locally Hölder-continuous of order α at a point τ our main result states that (formula present) converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation and the drift-function are of the type |t|^α.