Which of the following sets are convex?
Given sets are
• {(x,y) : x2 + y2≥ 1}
• {(x, y) :y2 ≥ x}
• {(x, y) : 3x2 + 4y2 ≥ 5}
• {(x, y) : y ≥ 2, y ≤ 4}
A convex set, is nothing but whose solution set is in the shape of a convex polygon.
If we map these functions on a graph, we can clearly find the set with a convex solution set.
• f : {(x,y) : x2 + y2≥ 1}
From the graph, it is evident that the solution set which is the shaded region is not convex.
• g:{(x, y) :y2 ≥ x}
From the graph, it is evident that the solution set which is the blue shaded region is not convex.
• h:{(x, y) : 3x2 + 4y2 ≥ 5}
From the graph, it is evident that the solution set which is the grey shaded region is not convex.
• p:{(x, y) : y ≥ 2, y ≤ 4}
From the graph, the dark blue shaded region between the two bright lines is a convex set.
Hence, the solution is option D.