I'm studying the article "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de vecteurs" (polar factorization and monotone rearrangement of vector-valued functions).
In this work one function is build which is expected to have some properties:
let $f$ be a function from some compact space S, $f:S \mapsto S$.
$f$ is supposed to be bounded, Rieman-integrable and a Borel function.
let $b$ a measurable and bounded function.
let $g(y)=\sup_x(y \cdot f(x)+b(x)-1/2\lVert f(x) \rVert^2)$ for $y \in S \bigcup f(S)$ (NB I'm unsure for $\bigcup$. It may be $\bigcap$).
The assertion is made that $g$ is convex, lipschitz continue and differentiable but no proof is given. I understand that it is most certainly an immediate result for an expert in the field, which I'm not.
How can these properties be proven?
I trivially started from the definition of convexity: $g(ty_0+(1-t)y_1) = \sup_x(ty_0 \cdot f(x)+(1-t)y_1 \cdot f(x)+b(x)-1/2\lVert f(x) \rVert^2)$ but I don't know how to handle that further, when taking the supremum.
I also gave a try with $S$ a closed set of $\mathbb{R}$, $b=0$ and $f$ differentiable. I've got two extremum for $y \cdot f(x)-1/2\lVert f(x) \rVert^2$: $f'(x)=0$ and $y=f(x)$. Setting $z=f(x)$, I can see trivially that the second one is a maximum, then $g(y)=1/2y^2$ which has the desired properties. But, so far, it does not give me more insight for the more general version.