An example to proper convex functions $f_1, f_2$ such that the infimal convolution of $f_1$ and $f_2$ is not proper.

Question

An example to proper convex functions $f_1, f_2$ such that the infimal convolution of $f_1$ and $f_2$ is not proper.

I need to find a function $g(x) = \inf\{f_1(x-y) + f_2(y)| y \in \mathbb{R}^n\}$ where $f_1$ and $f_2$ are proper convex functions but I am not even sure where to begin.

Answer 1

A constructive approach is to find closed proper convex functions $f_1^*$ and $f_2^*$ such that their sum $f_1^*+f_2^*$ is not proper convex, and then to take the conjugates of $f_1^*$ and $f_2^*$. Such functions are easy to find by letting $f_2^*$ take the value $\infty$ whenever $f_1^*$ is finite. This uses the fact that $f^{**}=f$ for closed proper convex functions and that $(f_1^*+f_2^*)^*$ is the infimal convolution of $f_1$ and $f_2$.

Answer 2

Hint: take $f_1 \equiv 42$ and a nice $f_2$.