What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?
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I think you should be able to do this for arbitrary $m$ by taking the points $0$, $e_1$,
$$\cos\left(\frac{2\pi}{m}\right) e_1 + \sin\left(\frac{2\pi}{m}\right) e_2,$$
and
$$\cos\left(\frac{2\pi}{m} k\right) e_1 + \sin\left(\frac{2\pi}{m} k\right) e_2 + e_{k+1}$$
for $2 \leq k \leq m -1$.
Subtracting 0 from all the other points, we get a bunch of vectors that are linearly independent -- note $\sin(2\pi/m)\neq 0$ for $m \geq 3$ -- so these points span an $m$-simplex in $\mathbb{R}^m$.
The projection of this simplex onto the first two components gives all affine combinations of the vertices of a regular $m$-gon and zero, which is a solid regular $m$-gon.
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$\begingroup$ Does this simplex admit a non-singular linear transformation to the simplex with corner points of shape $[11\dots1100\dots00]\in\{0,1\}^{n-1}$ (all $0$s and all $1$s also considered and so we have $n$ points of which $n-1$ are linearly independent)? If the simplex vertices in your polygon also form $n-1$ independent vectors then the linear transformation should exist. $\endgroup$– VS.Commented Apr 16, 2019 at 10:21