Rounding on the standard simplex: regular grids for global optimization

Given a point on the standard simplex, we calculate a proximal point on the regular grid which is closest with respect to any norm in a large class, including all `-norms for p ≥ 1. We show that the minimal `-distance to the regular grid on the standard simplex can exceed one, even for very fine mesh sizes in high dimensions. Furthermore, for p = 1, the maximum minimal distance approaches the `-diameter of the standard simplex. We also put our results into perspective with respect to the literature on approximating global optimization problems over the standard simplex by means of the regular grid.