Products of convex functions need not be convex
= Products of convex functions need not be convex
The <affine functions> $x$ and $-x$ are both <convex functions> on $\mathbb R$, but their product is $-x^2$. Its value at zero exceeds the average of its values at $-1$ and $1$, contradicting <convexity>. Thus closure under nonnegative linear combinations does not imply closure under products.