I want to construct an $n$-simplex the following way: - Choose $n$ vectors in the boundary of an $n$ dimensional ball, which are forming an $(n-1)$-simplex together. - Place the orthogonal affine $n-1$-hyperplane on each of these vectors. My question now is: Does the part enclosed by these hyperplanes together with the $(n-1)$-simplex now form an $n$-simplex?