What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n1$ dimensions whose projection yields boundary of a regular $m$gon on $2D$ plane?
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.

$\begingroup$ Does this simplex admit a nonsingular linear transformation to the simplex with corner points of shape $[11\dots1100\dots00]\in\{0,1\}^{n1}$ (all $0$s and all $1$s also considered and so we have $n$ points of which $n1$ are linearly independent)? If the simplex vertices in your polygon also form $n1$ independent vectors then the linear transformation should exist. $\endgroup$ – VS. Apr 16 at 10:21