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I'm looking for $x_1,x_2,x_3$ which satisfy the following constraints:

$$ \begin{align*} &x_1,x_2,x_3\geq 0\\ &x_1+x_2\geq a\\ &x_2+x_3\geq b\\ &x_3+x_1\geq c\\ &x_1+x_2+x_3=1 \end{align*} $$

where $a,b,c\geq 0$.

It is easy to find a necessary condition: $$ \begin{align*} &a+b+c\leq2\\ &0\le a,b,c\leq1 \end{align*} $$

I was told that it is also a sufficient condition. However, I cannot prove it. Can anyone help me?

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