Prove that a set in $\Bbb R^3$ is not convex

Question

Any tips how to prove that the set

$$S = \{ x \in \mathbb{R^3} \mid x_1-x_2^2 \le x_3 \le x_1+x_2^2\}$$

is not convex? Any hint how to start would be very helpful.

Answer 1

You can check that if $(x_1, x_2, x_3) \in S$, then $(x_1, -x_2, x_3) \in S$ using definition of $S$.

Thus $(x_1, 0, x_3) \in S$ if $S$ is convex and $(x_1, x_2, x_3) \in S$. This is because $(x_1, 0, x_3)$ lies on the line segment connecting $(x_1, x_2, x_3)$ and $(x_1, -x_2, x_3)$.

Note that $(1, 1, 0) \in S$ but $(1, 0, 0) \notin S$. Hence $S$ is not convex.