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Given the regular $n$-dimensional simplex $S\subset\Bbb R^n$ with $n\ge 4$, as well as two orthogonal subspaces $V,W\subset\Bbb R^n$ of dimension $\ge2$ (not necessarily of same dimension, not necessarily $V\oplus W=\Bbb R^n$).

We consider orthogonal projections of $S$ onto these subspaces, denoted by $S_V$ (resp. $S_W$). A face of $S$ is said to be exposed in a projection, if it gets mapped onto the boundary of $S_V$ (resp. $S_W$).

Question: Can it happen that $S_V$ and $S_W$ expose all 0-faces (vertices) of $S$, and expose the exact same 1-faces (edges) of $S$?

Especially interesting are examples in which the projections are not neighborly (i.e. not all edges are exposed).


Notes

  • The question what sets of edges of $S$ can be exposed is equivalent to the question what graphs are the 1-skeletons of polytopes $-$ a famously unsolved problem. This question only asks if a set of edges, if exposable at all, can possibly be exposed by different projections to orthogonal subspaces.

  • If $V$ and $W$ are not required to be orthogonal, then this is obviously possible. We can even require the projections $S_V$ and $S_W$ to be not combinatorially equivalent: take two non-equivalent neighborly polytopes with $n$ vertices and represent them as projections of the same $(n-1)$-dimensional simplex.

  • It is not sufficient that the projections are combiantorially equivalent. They have to use the exact same 1-faces of $S$. I am not sure whether this is a restriction since we might be able to apply a symmetry transformation of $S$ to one of the subspaces to make the exposed 1-faces line up. However, it is not clear that this preserves the orthogonality of $V$ and $W$.

  • I am aware of the paper [1], which shows that for $n\to\infty$ the projection of $S\subset\Bbb R^n$ onto a randomly chosen $d$-dimensional subspace with $d=\lfloor \delta n\rfloor,\delta\in(0,1)$ is neighborly with probability converging to one. This seems to make it very likely that such two spaces $V$ and $W$ might exist. However, this is non-obvious and an explicit example would be welcome. It is because of this paper that I consider it more interesting when the projections are not neighborly. In higher dimensions it seems to be unlikely that a certain (non-complete) set of 1-faces is exposed, therefore even more unlikely that two projections onto orthogonal spaces expose the same 1-faces.


[1] Donoho, Tanner: Neighborliness of randomly projected simplices in high dimensions

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