Moreau’s Decomposition Theorem Revisited

Abstract

Given two convex functions $\mathrm{g}$ and $\mathrm{h}$ 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 $\mathrm{F}$ 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 $\mathrm{F}$ is given as a deconvolution of a convex function by another one. The approach taken here as well as the way of factorizing $\mathrm{g}$ and $\mathrm{h}$ shed a new light on what is known as Moreau’s theorem in the literature on Convex Analysis.