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 $\ell^p$-norms for $p\ge 1$. We show that the minimal $\ell^p$-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 $\ell^1$-diameter of the standard simplex. We also put our results into perspective to the literature on approximating global optimization problems over the standard simplex by means of the regular grid.
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