Denote the class of lower-semi-continuous convex functions $f:\mathbb{R}^n\to \mathbb{R}\cup\{\pm\infty\}$ by $Lscx(\mathbb{R}^n)$ ( so that only function attaining the value $-\infty$ is the constant $-\infty$ function). Denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the standard scalar product on $\mathbb{R}^n$.
The classical Legendre transform $\mathcal{L}:Lscx(\mathbb{R}^n)\to Lscx(\mathbb{R}^n) $ is given by $$ \mathcal{L}[f](u)=\sup_x(\langle x,u \rangle -f(u)). $$
And this transformation is well defined (see here p.12) and $$ \begin{array}{rcl} \mathcal{L}\circ\mathcal{L}[f]&=&f,\\ f\leq g &\implies& \mathcal{L}[f]\leq \mathcal{L}[g] \end{array} $$ see here theorem 1.
My Question: Supose that for all $x\in\mathbb{R}^n$ we have $-F(x,\,\cdot\,)\in Lscx(\mathbb{R}^n)$ and for all $y\in\mathbb{R}^n$ we have $F(\,\cdot\,,y)\in Lscx(\mathbb{R}^n)$. Under these assumptions the statement $$ \mathcal{L}[F(\,\cdot\,,z)](x) = \mathcal{L}[-F(w,\,\cdot\,)](y) \;\;\forall z,w\in\mathbb{R}^n \implies \max_v\min_u F(u,v)=\min_u\max_vF(u,v)\; $$ is true?
I'm trying to use theorem 1 and lemmas 1, 2 and 3 of this Article.