given the following 3 sets:
$ \{ (x,y,z): x \ge y^2 + z^2, z>0 \} $
$ \{ (x,y,z): x^2 \ge y^2 + z^2, y>0 \} $
$ \{ (x,y,z): x^2 \ge y^2 + z^2, x>0 \} $
The first set is convex because it is a sum of convex set.
Why is the second set not convex? I thought it is also a convex sum: $y^2+z^2-x^2$? And why is in example 3: $-x^2+y^2+z^2$ again a convex sum?
Thank you very much for your help!
For set #2: $(−1,1,0)$ belongs and $(1,1,0)$ belongs but convex combination $(0,1,0)$ doesn't belong. Note that $-x^2$ isn't a convex function and $y^2+z^2-x^2$ is likewise not convex.
For set #3: If $x > 0$ then: $$ x^2 \geq y^2 + z^2 \Leftrightarrow 0 \geq \sqrt{y^2 +z^2} - x $$ Observe $f(x,y,z) = \sqrt{y^2+z^2} - x$ is a convex function since it's the sum of a convex function (2-norm of (x,y)) and an affine function ($-x$). The sub-level set of a convex function is a convex set.