Consider $(f\Box g) (z)=\inf_{x+y=z}(f(x)+f(y))$

Question

Consider $$(f\Box g) (z)=\inf_{x+y=z}(f(x)+g(y))$$

Find two convex functions $f$ and $g$ from $X$ to $\mathbb{R}$ such that the infimum given above is never a minimum.

Can anyone help with finding such functions?

Answer 1

A simple example would be $f(x) = x$, $g(x) = -x$: $$ (f\Box g) (z) = \inf_{x \in \mathbb R} (f(x) + g(z-x)) = \inf_{x \in \mathbb R} (2x - z) = - \infty \, . $$

$f(x) = e^x$, $g(x) = e^{-x}$ is an example where the infimum is finite: $$ (f\Box g) (z) = \inf_{x \in \mathbb R} (f(x) + g(z-x)) = \inf_{x \in \mathbb R} e^x (1 + e^{-z}) = 0 $$ and the infimum is not a minimum because $e^x (1 + e^{-z}) > 0$ for all $x$ and $z$.

Slightly more general, you can take any convex function $f :\mathbb{R} \to \mathbb{R}$ which does not have a minimum, and define $g$ by $g(x) = f(-x)$.