I have been given the following task: Let $P \subseteq \mathbb{R}^d$, a polytope, such that $P = \text{conv}(K)$ for $K \subseteq S^{d-1}$ finite, where $S^{d-1}$ denotes the Euclidean unit sphere in $\mathbb{R}^d$. Furthermore, assume that $0 \in P$ and that $K$ contains $d$ vectors that are linearly independent. Show that $\text{Aut}(P)$ is finite.
I am wondering about all the conditions I have given, because I thought that I would only need $d+1$ affinely independent points and the information that I have finitely many vertices to conclude that the autonorphism group is finite. Why do i need $0 \in P$ and why all the stuff with $S^{d-1}$. Thank you in advance.