Convex Analysis is devoted to study convex sets, convex functions and related extremal problems (minimization, maximization) in finite and infinite dimensional spaces. The course will remain mainly in the framework of normed spaces. I shall treat some of the topics listed below (the choice will depend on knowledge and interests of the students), following one of the two lines: "general line" or "functional line".
Expected learning outcomes
Knowledge of the topics of the course and their application to simple theoretical problems.
· Convexity in vector spaces: algebraic separation theorems. · Short account of topological vector spaces: weak topologies. · Convexity in normed and Banach spaces: topological properties of convex sets, convex hulls, continuity of convex functions, topoloigical speration theorems, extendability of convex Lipschitz functions. · Finite dimensional convexity: relative interior, Carathéodory theorem and Helly theorem, Jensens's theorem and Hermite-Hadamard theorem, extreme ponts of finite dimensional convex sets (Minkowski thm.). · Minimization of convex functions.
General line o Theorems by Krein-Milman and Milman on representability of convex sets by their extreme points, Bauer' maximum principle. o Boundaries and support points: Bishop-Phelps theorem, theorems of Rodé and James (separable case). o Duality of convex sets: anihilators and polars. o Convex series, CS-closed and CS-compact sets. o Subdifferential of convex functions and differentiability. o Differentiability of convex functions outside small sets, short account of Asplund spaces. o Selections of multivalued mapping with convex values, Michael's theorem.
Functional line Subdifferential and subdifferential calculus. Short account on subdifferential and differentiability. Ekeland's variational principle, Bronsted-Rockafellar theorem. Fenchel duality. Aproximation of convex functions through infimal convolutions. Minimax theorems.