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I am wondering if there exists a surjective affine mapping from the n-cube to a regular convex polygon with k^n vertices (for any k? maybe just some k?). As an example I tried to think of a surjective mapping from the square to a regular convex polygon with 3^2 vertices.

Any ideas how to approach this?

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  • $\begingroup$ I think I'm missing something in your question. Can't you just put a regular k^n-gon K inside the n-cube and let f(x) be the nearest point in K to x? $\endgroup$
    – Yoav Kallus
    Commented Mar 3, 2021 at 16:25
  • $\begingroup$ Should the mapping be affine? $\endgroup$
    – Guillaume Aubrun
    Commented Mar 3, 2021 at 16:37
  • $\begingroup$ Yes, sorry I forgot to mention that it should be affine. $\endgroup$
    – timudk
    Commented Mar 3, 2021 at 17:17

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No such map exists.

A $k$-gon is the affine image of the $n$-dimensional cube iff it is centrally symmetric and $k/2 \leq n$. This is because the $n$-cube can be written as the Minkowski sum of $n$ segments in linearly independent directions, and therefore a planar set is an affine image of the $n$-dimensional cube iff it is the Minkowski sum of $n$ arbitrary segments in $\mathbf{R}^2$.

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