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Write the convex hull of the points $(0,0), (1,1), (2,0) \in \mathbb R^2$ as the set of solutions to a finite number of inequalities.

The convex hull of $x_1, \ldots, x_n \in \mathbb R^n$ is defined as the set $\{\lambda_1x_1 + \ldots + \lambda_n x_n : \sum_{i=1}^{n} \lambda_n = 1, \lambda_i \ge 0\}$.

I see that the convex hull of our points is $X = \{\lambda_1 (0,0) + \lambda_2 (1,1) + \lambda_3 (2,0) : \lambda_1 + \lambda_2 + \lambda_3 = 1, \lambda_i \ge 0\}$ however I've had no luck in expressing this set as a finite number of inequalities. How is this done ?

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You just have to draw a diagram of the region. You will see that it is the intersection of three half-planes. For instance, one of the half-planes, corresponding to the line from $(0,0)$ to $(1,1)$, contains all points $(x,y)$ such that $x \ge y$: this is one of your inequalities. I will let you find the other two.

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  • $\begingroup$ Is this formal enough ? I mean, how do I see every point in this region lies in convex hull ? $\endgroup$
    – Shuzheng
    Sep 6 '14 at 9:55

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