On the displacement function of isometries of Euclidean buildings

In this note we study the displacement function dg(x):=d(gx,x) of an isometry g of a Euclidean building. We give a lower bound for dg(x) depending on the distance from x to the minimal set of g. This answers a question of Rousseau (2001, 4.8) and Rapoport-Zink (1999, 2.2).