Let $A$ be a linear $m$-dimensional subspace of $\mathbf{R}^n$ $m < n$, containing the point $(1,1,\ldots,1) \in \mathbf{R}^n$, and consider the intersection of $A$ and the unit cube $\Delta_n$ (centered at the origin).
I'm interested in the behavior of $\Delta_n \cap A$. Can this body only be a parallelotope, or are there counter-examples?