# Given convex l.s.c. function $f$, find decreasing convex function $\phi$ such that $f(x) \equiv \sup_y x\phi(y)-\phi(-y)$

Let $$f: \mathbb R \rightarrow (-\infty,+\infty]$$ be a lower-semicontinuous convex function.

# Question

• Under what futher conditions does there exists a convex decreasing function $$\phi: \mathbb R \rightarrow \mathbb R$$ such that $$f(x) = \sup_y x\phi(y)-\phi(-y)$$ for all $$x \in \mathbb R$$ ?
• Construct such a $$\phi$$ explicitly.

# Examples

$$f(x) = -2\sqrt{x}$$ if $$x \ge 0$$ and $$f(x)=+\infty$$ else, then one may take $$\phi(x) \equiv e^{-x}$$.

# Observations

Given a decreasing function $$\phi$$, define its (pseudo-)inverse by $$\phi^{-1}(x):=\inf\{t \in \mathbb R \mid \phi(t) \le x\}$$, with the usual convention that $$\inf\emptyset = +\infty$$. Let $$T_\phi(x) := \phi(-\phi^{-1}(x))$$, for all $$x \in \mathbb R$$. Then, with the change of variable $$z := \phi(y)$$ one has

$$\sup_y x\phi(y)-\phi(-y)=\sup_zxz-\phi(-\phi^{-1}(z)) := T_\phi^*(x),$$ where $$T_\phi^*$$ is the convex conjugate of $$T_\phi$$. This suggests that one should consider the following problem rather:

Reformulated problem. Given a convex l.s.c function $$g$$, find decreasing convex function $$\phi$$ such that $$g = T_\phi$$.

## This question has an open bounty worth +50 reputation from dohmatob ending in 4 days.

Looking for an answer drawing from credible and/or official sources.