Given an ordered set of basis vectors $B \subset \mathbb{R}^2$ with arbitrary order. Apply any permutation $\sigma_i(B)$, it results in order of the basis. And I can cumulatively sum the basis with such order to form a set of vertices $V_i$.
For example, $B = \{ b_1, b_2, b_3 \}$, if $\sigma_1 = (1, 3, 2)$ Then $V_1 = \{b_1, b_1+b_3, b_1+b_3 + b_2 \}$. If $\sigma_2 = (1, 2, 3)$ then $V_2 = \{b_1, b_1+b_2, b_1+b_2+b_3 \}$.
Given $B$ and a set of permutations $S = \{ \sigma_1, \sigma_2, \cdots, \sigma_n\}$, how can one find the vertices of convex hull of $\cup_{i=1}^n V_i$ ?
A naive approach and be simply apply all the permutations to $B$ and then compute the convex hull of all possible points. But I'm wondering if there is any efficient way to do so?