Moreau’s Decomposition Theorem Revisited
Jean‐Baptiste Hiriart‐Urruty, Ph. Plazanet · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 1989
Given two convex functions \mathrm{g} and \mathrm{h} on a Hilbert space, verifying \mathrm{g} + \mathrm{h} = \frac{1}{2}\left\|.\right\|^{2} , we show there necessarily exists a lower‐semicontinuous convex function \mathrm{F} such that \mathrm{g} = \mathrm{F}\:□\:\frac{1}{2}\left\|.\right\|^{2} and \mathrm{h} = \mathrm{F}^*□\:\frac{1}{2}\left\|.\right\|^{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.