Discussion
Which of the following sets are convex sets ?
(i) \(\{(x, y) \mid y = x\}\)
(ii) \(\{(x, y) \mid y \le |x|\}\)
(iii) \(\{(x, y) \mid y \ge x^{2}\}\)
Answer: C
The convex sets are (i) and (iii) only.
(i) The line \(y = x\) is convex: any segment between two of its points lies on it. (iii) The region above the parabola, \(y \ge x^{2}\), is the epigraph of the convex function \(x^{2}\), so it is convex.
(ii) is not convex. The points \((-1, 1)\) and \((1, 1)\) satisfy \(y \le |x|\), but their midpoint \((0, 1)\) does not, since \(1 \gt 0\). The region below the V-shaped graph of \(|x|\) has a notch at the origin. So every option containing (ii) is wrong, and '(i) only' leaves out (iii).
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