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
\[ ({\mathcal T}\varphi)=({\mathcal L}\varphi)(Bx+v_0)+\langle x,y_0 \rangle +c_0. \]


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