Definition of Concave set

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We know that a set is convex if the straight line joining any two points of the set lies completely in the set. Or, mathematically, a set $X$ is convex if

$$x_1, x_2 \in X, 0 \leq \lambda \leq 1 \Longrightarrow\lambda x_1+(1-\lambda)x_2 \in X$$

But how do we define a concave set? Can you give an example?

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2 Answers

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There is no such thing as a concave set, since the definition of the convexity of a set is different from the convexity of a function.

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A set is concave if and only if its complement is convex.

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