How to prove that the following set is not convex?
$$M = \left\{ \mathbb{R}^{3}: x_{1}x_{2}x_{3}\le 1,x_{1}+x_{3}\ge 2,x_{1} \ge 0 \right\}$$
Thanks for any help.
I tried to write it down as intersection of two sets $\{x_1x_2x_3 \le 1\}$ and $\{x_1+x_3 \ge 2,x_1 \ge 0\}$. The second set is convex (it is clear) and the first set is not convex. I have used theorem about level sets, the function $x_1x_2x_3$ is not convex so the level set is not convex. This is problem, it does not mean that the intersection is not convex
Guide:
It's important to analyze where is the "weakness" of a problem. The last two constraints are linear. If all the constraints are convex, then the intersection must be convex. The first constraint is the weakness, attack it.
Set $x_3=1$, now visualize the projected set. Prove that the projected set is not convex by constructing a pair of points in the set where its midpoint is not in the set.
It might help if you to visualize the set $xy \le 1$. Give it a try.
Please try the following question first:
Prove that $M_2 = \{ \mathbb{R}^2: x_1 x_2 \le 1, x_1 +1 \ge 2, x_1 \ge 0\}$ is not convex.