Is the Legendre transform connected to identity in any sense

Question

Is the Legendre transform connected to identity in any sense? Is there a "good" way to interpolate continuously between a convex function and its Legendre transform in the sense that

$\gamma: [0,1] \rightarrow \{ \text{mappings of convex functions into convex functions} \}$

such that

$\gamma(0) = id \quad \text{and} \quad \gamma(1) = \text{Legendre transform}$

Answer 1

Yes, this is possible using the proximal average. The link to the official paper is https://epubs.siam.org/doi/10.1137/060664513

Answer 2

I doubt there's anything like what you're seeking. Let $X$ be a Banach space, for a proper function $f:X\to \Bbb R\cup \{\infty\}$, its Legendre-Fenchel transform $f^*:X^*\to\Bbb R\cup\{\infty\}$ is defined on a totally different space.

Given that you want $\gamma$ such that $\gamma(0)=f$ and $\gamma(1)=f^*$, the domain of the function $\gamma(t)$ for $t\in (0,1)$ is already a problem. Do you want it to be $X,X^*$ or something else entirely? You'd need to think of a continuous way to transform $X$ into $X^*$ (and I'm not even sure how to make that precise).