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Let $(x_1,y_1),\dots,(x_n,y_n)$ be a set of points in $\mathbb{R}^2$. Define the variables $z_i=x_i-\lambda y_i$. For any real number $\lambda$, I define the set $Z^{+}(\lambda)$ of numbers $$Z^{+}(\lambda) = \{z_i\mid z_i \geq 0\}$$ Thus $Z^{+}$ contains the set of positive numbers among $z_i$ for a given $\lambda$. Now consider the function \begin{align}f(\lambda)=\lvert Z^{+}(\lambda)\rvert\end{align} where $\lvert . \rvert$ is cardinality of the argument set. Is $f(\lambda)$ concave?

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This function is not continuous, and each convex/concave function is continuous, so this function is not convex.

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