Slopes and Moreau-Rockafellar Theorem

TL;DR

The paper develops a slope-centric framework for analyzing functions on complete metric spaces, introducing local and global slopes $|\nabla f|$ and $|\widetilde{\nabla} f|$, and exploring their properties and critical sets $0$-crit and $\varepsilon$-crit. It proves primal results linking the infimum of $f-g$ to minimizers on critical sets, leveraging Ekeland's variational principle and slope invariance under restricted domains. Building on these tools, it yields a primitive, non-dual proof of the Moreau-Rockafellar theorem in convex analysis: if $\partial g\subset\partial f$ then $f=g+c$ on the Banach space. The work bridges variational analysis in metric spaces with classical convex duality, offering a unified method that underpins gradient-flow analysis and convex equality results without transfinite constructions.

Abstract

Properties of local and global slope of a function and its approximate critical points sets are studied in relation to determination of the function.

Theorems & Definitions (36)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof