Prove a set is convex algebraically

**MSUMathStdnt** · January 30th 2011, 10:32 AM

How can I determine if a set is convex without graphing it?

I have that a set, $S$ , is convex if, given any two points, $x_1, x_2 \in S$ , the line connecting them is entirely within $S$ , or:

$\lambda x_1 + (1-\lambda) x_2 \in S \qquad \forall \lambda \in [0,1]$

Some of the sets in question:
b. $\{(x,y,z) \colon z=\left| y \right|, x \le 3 \}$
c. $\{(x,y): y-3x^2=0\}$
d. $\{(x,y,z): y \ge x^2, x+2y+z \le 4\}$
e. $\{(x,y):x=3, \left| y \right|, \le 4 \}$
f. $\{(x,y,z): z= \left| y \right|, x \le 3 \}$

I have to hand in f. I'm not good enough to draw it, but I can envision it. It looks like a "V" going from $x=3$ down to $-\infty$ . Clearly, it is not convex. But how would I show that mathematically?

Thanks.