Question

Let $S_1 = {(x, y) ∈ R^2 ∶ x + y ≥ 1, x + y ≤ 2, y ≥ x^2, x, y ≥ 0}$ and $S_2 = {(x, y) ∈ R^2 ∶ x + y ≥ 1, x + y ≤ 2, y ≤ x^2, x, y ≥ 0}.$ Then, which of the following is CORRECT ?

• A. Both $S_1$ and $S_2$ are convex sets
• B. $S_1$ is a convex set but $S_2$ is not a convex set
• C. $S_1$ is a convex set but $S_2$ is not a convex set
• D. Neither $S_1$ nor $S_2$ are convex sets

Hint:

170

Answer 1

The Correct Option is B

Convexity Analysis of Regions S₁ and S₂ 

The convexity of a set is determined by whether the line segment between any two points in the set remains entirely within the set.

Region S₁: y ≥ x²

The region S₁ is defined by the constraint:

y ≥ x²

This means that S₁ includes all points above the parabola y = x². Despite the presence of x², which often suggests non-convexity, this region remains convex because any line segment drawn between two points within S₁ remains within the set.

Region S₂: y ≤ x²

The region S₂ is defined by the constraint:

y ≤ x²

Here, S₂ includes all points below the parabola y = x². However, this region is non-convex because the parabolic curve y = x² creates gaps where line segments between two points in S₂ can extend outside the region.

Conclusion:

• S₁ (y ≥ x²) is convex since it includes all line segments between any two points within the set.
• S₂ (y ≤ x²) is non-convex because certain line segments between points in S₂ extend outside the region.