Let $\Gamma$ be a cocompact lattice in $\mathbb{R}^n$, eg. $\Gamma = A \mathbb{Z}^n$ for some $A \in SL_n \mathbb{R}$. Then any $k$-dimensional subspace $P$ which is rational in $\Gamma$ has a volume: ie. $P$ rational in $\Gamma$ means $P \cap \Gamma$ is a cocompact lattice in $P$ and we assign $vol(P, \Gamma)=covolume(P\cap \Gamma / P)$. Concretely if $x_1, \ldots, x_k$ is a $\mathbb{Z}$-basis for $P \cap \Gamma$, then $vol(P, \Gamma)^2=det(\langle x_i, x_j \rangle)$.
Now there is the following basic principle concerning minimal volume rational 1-dimensional subspaces of $\Gamma$ (ie. the shortest nonzero vectors of $\Gamma$, or $syst_1 (\Gamma)$): if $x, x'$ are distinct nonzero shortest lattice elements in $\Gamma$, then the angle between $x,x'$ is 'large'. This is just a computation: for if the angle between $x,x'$ is sufficiently small, then their difference $x-x'$ will be a shortER lattice element.
My question: is there an analogous principle for minimal volume rational $k$-dimensional subspaces in $\Gamma$? More specifically, suppose we consider the $\mathbb{R}$-span $W$ of the minimal volume $k$-dimensional subspaces in $\Gamma$. If $W \neq \mathbb{R}^n$, then there will exist a minimal volume rational $k$-plane $P$ which is not contained in $W$. If we had $k=1$, then we could say that the projection of $P$ onto the orthogonal complement of $W$ is 'large' (ie. $P$ has definite angle with $W$). What is the proper analogous statement for arbitrary $k$?