$ V $-Moreau envelope of nonconvex functions on smooth Banach spaces

Abstract

<p>We continue the study of the properties of the $ V $-Moreau envelope and generalized $ (f, \lambda) $-projection that we started in <sup>[<xref ref-type="bibr" rid="b5">5</xref>]</sup>. In this paper, we study the differentiability and the regularity of the $ V $-Moreau envelope and the Hölder continuity of the generalized $ (f, \lambda) $-projection. Our results extend many existing results from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. Even on Hilbert spaces and for convex functions and sets, we derived new results.</p>