Moreau’s Decomposition Theorem Revisited

Abstract

Given two convex functions g and h on a Hilbert space, verifying $\mathrm{g}+\mathrm{h}=\frac{1}{2}{\Vert .\Vert }^2$, we show there necessarily exists a lower‐semicontinuous convex function F such that $\mathrm{g}=\mathrm{F}\square \frac{1}{2}{\Vert .\Vert }^2$ and $\mathrm{h}={\mathrm{F}}^{\ast }\square \frac{1}{2}{\Vert .\Vert }^2$. An explicit formulation of F is given as a deconvolution of a convex function by another one. The approach taken here as well as the way of factorizing g and h shed a new light on what is known as Moreau’s theorem in the literature on Convex Analysis.