Prove - Convex set

[pidaro]

Prove that the set S = {(x, y) ∈ R^2: x + 2y = 4} is a convex set.

 

[pka]

Prove that the set S = {(x, y) ∈ R^2: x + 2y = 4} is a convex set.

Tell us what it means to say that the set \(\bf S\) is convex.

 

[HallsofIvy]

Suppose \(\displaystyle (x_1, y_1)\) and \(\displaystyle (x_2, y_2)\) are two points in this set. That is
\(\displaystyle x_1+2y_1= 4\) and \(\displaystyle x_2+ 2y_2= 4\).

To show that the set is convex you must show that any point on the line segment between the two points, which can be written \(\displaystyle (ax_1+ (1-a)x_2, ay_1+ (1-a)x_2)\) for a between 0 and 1, is also in that set. That is, that \(\displaystyle ax_1+ (1-a)x_2+ 2(ay_1+ (1-a)x_2)= 4\) also.