I need to prove this (assuming it's true):
The vector sum of the vectors pointing to the vertices of an n-sided regular polytope whose center is at the origin of a Euclidean space is zero.
If it has an even number of vertices, it's clearly zero by symmetry (at least where my imagination works). I can't think of a way to prove this in the general case, though.
Best Answer
The Chebyshev center of a regular polytope is unique. Assume that the Chebyshev center $C$ of a regular polytope with centroid in the origin is not the origin. Then we must have $\varphi(C)=C$ for any $\varphi$ in the symmetry group of the polytope, or $C\in\operatorname{Fix}(\varphi)$. If we have two different elements in the symmetry group of the polytope such that the corresponding sets of fixed points are lines through the origin, we have a contradiction. But the last condition is met by any polytope with more than two faces, hence it is trivial.