Given a set of vectors $\mathbf v_i$ for $i=1,\dots,k$, $\mathbf v_i \in \{0,1\}^n$, is that possible to efficiently find the volume of the set,
$\left\{\mathbf x \in [0,1]^n:\mathbf x \le \sum_{i=1}^k \alpha_i\mathbf v_i\ \text{for some $\alpha_i$}\right\},$ such that $\sum_i \alpha_i=1$ and $\alpha_i \ge 0$. The comparison $\mathbf x \le \sum_{i=1}^k \alpha_i\mathbf v_i$ is taken componentwise.
Note: the above question arose from another question asked by me previously.