There are a number of operations that may be done for convex functions such that the resulting function is convex as well.

What about the opposite? Do the operations that preserve convexity also preserve non-convexity?

In particular, I am interested whether the following two statements are true:

  1. Vector composition: Let $g_i(x)\in\mathbb{R},i=1,\dots,n$ be convex and $h(y)\in\mathbb{R},y\in\mathbb{R}^n$ be non-convex. Then, $h(g_1(x),\dots,g_n(x))$ is non-convex.

  2. Positive weighted sum: Let $h_j(y)$ be non-convex. Then $\sum_{j=1}^m \alpha_j h_j(y),\alpha_j>0,\sum_{j=1}^m\alpha_j=1$ is non-convex.


(1) fails for $g(x) = x^4$ and $h(x) = \sqrt(|x|)$.

(2) fails for $h_1(x)=\sin(x)$, $h_2(x)=-\sin(x)$, $\alpha_1=\alpha_2=\frac12$.


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