Is this a convext set?

Question

Is this one a convex set? how to prove it? I failed to prove it through the definition of convex set. Thank you.

$$\left\{(x_1,x_2)\mid\sqrt{x_1^2+x_2^2}+|x_1|+|x_2|\le 1\right\}$$

score 3 · Accepted Answer · answered Sep 28 '13 at 06:29

3

Any norm is convex, and a sum of convex functions is convex, so $f(x) = \|x\|_2 + \|x\|_1$ is convex. It follows that $\{ x \mid f(x) \leq 1\}$ is convex.

answered Sep 28 '13 at 06:29

littleO

51,938

Thank you! I have also succeeded in proving it through the definition of a convex set. But your method is much cleverer! – happyWinner Sep 28 '13 at 09:14

Is this a convext set?

1 Answers1