Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How can we check the convexity of a set ?

$S = \{(x,y) \in \mathbb{R}: x-y²\le y \le x+y²\}$

share|improve this question
What have you tried... dot com? –  kahen Nov 26 '12 at 13:21

1 Answer 1

up vote 0 down vote accepted

A set is $S \subseteq \mathbb R^2$ is convex iff $(x_1, y_1), (x_2, y_2) \in S$, $\lambda \in [0,1]$ imply that $\bigl((1-\lambda)x_1 + \lambda x_2, (1-\lambda)y_1 + \lambda y_2\bigr) \in S$. Check this for your set $S$, that is, check whether $$ x_i - y_i^2 \le y_i \le x_i + y_i^2, \qquad i \in \{1,2\} $$ imply $$ (1-\lambda)x_1 + \lambda x_2 - \bigl((1-\lambda)y_1 + \lambda y_2\bigr)^2 \le (1-\lambda)y_1 + \lambda y_2 \le (1-\lambda)x_1 + \lambda x_2 + \bigl((1-\lambda)y_1 + \lambda y_2\bigr)^2 $$ for each $\lambda \in [0,1]$.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.