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?