## The concept of duality in convex analysis, and the characterization of the Legendre transform.(English)Zbl 1173.26008

Let $$Cvx(\mathbb{R}^n)$$ be the class of lower semi-continuous convex functions $$\varphi:\mathbb{R}^n\to\overline\mathbb{R}$$ (such that the only function attaining the value $$-\infty$$ is the constant $$-\infty$$ function) and let $${\mathcal L}:Cvx (\mathbb{R}^n)\to Cvx(\mathbb{R}^n)$$ be the Young-Fenchel transform according to
$({\mathcal L}\varphi)(x)=\sup_y(\langle x,y\rangle-\varphi(y)).$
It is well known that $${\mathcal L}$$ is order-reserving (i.e. $$\varphi\leq\psi$$ implies $${\mathcal L}\varphi\geq {\mathcal L}\psi)$$ and that $${\mathcal L}$$ is an involution (i.e. it is $${\mathcal L}{\mathcal L}\varphi=\varphi$$ for any $$\varphi\in Cvx(\mathbb{R}^n))$$.
In the paper, the authors show conversely that any involution $${\mathcal T}:Cvx(\mathbb{R}^n)\to Cvx(\mathbb{R}^n)$$ which is order-reversing must be, up to linear terms, the Young-Fenchel transform, i.e. there exist a invertible symmetric linear transformation $$B: \mathbb{R}^n\to\mathbb{R}^n$$, a vector $$v_0\in\mathbb{R}^n$$ and a constant $$c_0$$ such that
$({\mathcal T}\varphi)=({\mathcal L}\varphi)(Bx+v_0)+\langle x,y_0 \rangle +c_0.$

### MSC:

 26B25 Convexity of real functions of several variables, generalizations 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces)
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