Sums of n convex functions

Question

Good Night, I have a question about the sums of convex functions, for example when we have the sum of two convex functions, the minimum of the resulting of convex function always lies between the minimum of each convex functions?. This is applicable to the sum of "n" convex functions? the minimum of the resulting function exists in the area defined by the minima of each function? Could you recommend a bibliographical reference where I could read about this generalization?. Thanks.

Answer 1

The minimum of the sum lay in the convex hull of the minima of all the separate functions when we are in a one dimensional space. To see that, you can observe that since the convex function $f_i$ have minimum $x_i$, for $i\in[n]$, then it must be decreasing for $x<x_i$ and increasing for $x>x_i$. Suppose that $x<\min_i x_i$, then for all $i\in[n]$, $f_i(x)\geq f_i(\min_i x_i)$, the same happens if $x>\max_i x_i$.