# Extreme points from constraint expression of convex space

I'm looking for the extreme points of the convex set $S\subset [-1,1]^{n\times 3}$ with $r\in S$ such that $$r_{i} \ge r_{k} \iff i\ge k,$$ where the first inequality refers to lexicographical inequality in $\mathbb{R}^3$, and $$\sum_i^n r_i = 0$$ It was easy to characterize the extreme points satisfying only the first set of constraints. The second set has proven to be more difficult and I don't want to waste my time when there's already a solution out there.

Are there any program packages or methods out there that would easily solve this problem? If so, what are they?

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The algorithm runtime will be exponential in the dimensionality of the polyhedron. To enumerate all of the vertices of an $n$-cube, all $2^n$ vertices must be enumerated. The problem is probably NP-hard.