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?



Projections of particular simplex yielding boundary of a regular polygon?
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.


