I am a theoretical computer science student who is not well versed in convex geometry, so could someone point me to the correct name of the following type of convex polytopes (if they are named)?
A convex polytope $P$ which is symmetric with respect to $S_n$ group action. Formally, let $\bar{v} \in \mathbb{R}^n$ be a vector. Then, $\bar{v} \in P$ if and only if $\sigma\circ \bar{v} \in P$, for every $\sigma \in S_n$, where $S_n$ is the finite symmetric group. Here, $\sigma\circ \bar{v}$ represents a permutation of the original vector $\bar{v}$.