Suppose in $\mathbf{R}^n$ there are $m$ given hyperplanes $\Pi_j:\sum_{i=1}^n c_{i,j}e_i=0$ all of which go through the origin, and all the coefficients $c_{i,j}$ are rational (you can make them all integers if you like). Clearly, each hyperplane contains some integer points near the origin.
Consider a moving hyperplane $\Pi$ also through the origin. It could be arbitrarily close to one of the given planes say $\Pi_1$ so that the nearest integer point $P$ on $\Pi_1$ is arbitrarily close to $\Pi$. However, intuitively it cannot be arbitrarily close to two or more planes, and so the closest integer point on the other planes $\Pi_2$ through $\Pi_m$ to $\Pi$ should have distance bounded away from zero.
My question is, is there an estimate on the lower bound say $L>0$ in terms of $c_{i,j}$'s such that no matter of the moving plane $\Pi$, all non-origin integer points on $m-1$ hyperplanes must have distances at least $L$ from $\Pi$?
