Minimum of a sum of positive functions is the sum of the minimums of the functions

Question

Let $f_1, \dots, f_n$ be positive functions from $\mathbb R^m \rightarrow \mathbb R$.

How do we show that $$\min_x \sum_{i=1}^n f_i(x) = \sum_{i=1}^n \min_x f_i(x)$$

Actually, I am not sure this is true. Maybe adding convexity of the functions helps ?

Answer 1

It is not true. Take $f_1,f_2\colon\mathbb R\longrightarrow\mathbb R$ defined by $f_1(x)=(x-1)^2$ and $f_2(x)=(x+1)^2$. Then $\min f_1+\min f_2=0$, but $\min(f_1+f_2)=2$.