Legendre transform concave function

Question

Let $f$ be a concave function and define

$f^*(y) := \inf_{x}(yx-f(x))$. Is this in any sense related to the Legendre transformation? -If yes, is $f^*$ also concave? Is this transformation invertible in any way?

What happens if we apply the ordinary Legendre transformation $f^{**}(y) = \sup_{x} (yx-f^*(x))$ to $f^{*}$. Is there any way to reconstruct $f^{*}$ from $f^{**}$?

Please, anything is unclear, let me know.

Answer 1

Let $g(x) = -f(x), p = -y$.
$inf_x (yx - f(x)) = inf_x (- px + g(x)) = -sup_x (px - g(x))$
So if $L(y, f)$ is common Legendre transform, then your operator is $M(y, f) = L(-y, -f)$.
Properties can be derived.