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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?

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    $\begingroup$ Can you state the definition of a parallelotope? $\endgroup$ Jul 6, 2014 at 23:36
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    $\begingroup$ ...and clarify whether the "unit cube $\Delta_n$" is has side-length $1$ or radius $1$? (In the former case, $\Delta_n = [-1/2, +1/2]^n$ and the point $(1,1,\ldots,1)$ is in the exterior; while in the latter, $\Delta_n = [-1, +1]^n$ and you're asking about the the intersection with a subspace that contains one of the vertices.) $\endgroup$ Jul 7, 2014 at 0:55

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Counterexample: for $(m,n)=(3,4)$ we can get a regular octahedron as the intersection of the tesseract $\Delta_4$ with the hyperplane $x_1+x_2 = x_3+x_4$. [It's easier to think of the equivalent but more symmetrical $x_1+x_2+x_3+x_4=0$, where the pairs of opposite vertices are $\pm(1,1,-1,-1)$, $\pm(1,-1,1,-1)$, and $\pm(1,-1,-1,1)$ (or half of those vectors if "$\Delta_4$" is meant to have side-length $1$ rather than $\ell^\infty$-radius $1$).]

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