A property of the minima of a sum of convex functions, take 2

Question

This is a follow-up to my previous question. Let $g_1(x), \ldots, g_k(x)$ be convex functions from $\mathbb{R}^n$ to $\mathbb{R}$, and lets assume that global minimum of each $g_i$ is unique and is achieved, denoting $$x_i = \arg \min_{x \in \mathbb{R}^n} g_i(x).$$ Suppose all $x_i$ lie in the box $[a,b]^n$. Is it true that the minima $$ g_1(x) + g_2(x) + \cdots + g_k(x)$$ will lie in the same box $[a,b]^n$?

Answer 1

No. Take the counterexample given to your previous question, or any counterexample actually. There is a hyperplane separating the minimum of the sum function from the minima of the individual functions. By rotating and shifting the coordinate system, you can make that hyperplane become parallel to a coordinate hyperplane, with all the $x_i$ having all components positive. Now you have a counterexample with $a=0$ and $b$ sufficiently large.