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Consider the set $S$ in $\mathbb R^3$ defined by the inequalities:

$x+y+z \ge 1$

$-x+y+z \le 1$

$x-y+z \le 1$

$x+y-z \le 1$

How can I show that $S$ is a $3$-simplex ? (Convex hull of $3 + 1$ affinely independent points).

I've been looking over a lot of theorems in my book of convex optimization, but I've found none that could be directly applied.

How can I by inspection find these affinely independent points ?

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The question is whether $S$ is a simplex. Obviously, it is not. The constraint set of 3-simplex in $R^3$ is $x+y+z=1$

$x,y,z>=0$

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