Funções convexas e as transformadas de Legendre e Fenchel

Abstract

The present work addresses the main elements of convex analysis in vector spaces of finite and infinite dimensions. In finite-dimension, it presents fundamental concepts of norms, inner-product, and topology. Then, it defines convex sets and explores their properties. It shows operations that preserve convexity, classic convex sets, and the hyperplane separation theorem. Next, the work presents the convex functions and their properties, from which we can highlight the continuity in open subsets and the existence of the directional derivative. The theoretical framework developed allows presenting the Legendre transform when the convex functions are C 1 and the Fenchel transform for non-smooth convex functions. Among all applications of the Legendre transform, this work highlights the formulation of equations of classical mechanics. A table with selected smooth convex functions and their respective Legendre transform is shown. In infinite dimension, the work develops topological concepts and properties of metric spaces, continuity, Bolzano-Weierstrass theorem, Hilbert and Banach spaces, and Hahn-Banach theorem. Then, it defines interior points, convex sets, and convex functions in Hilbert spaces, defining main properties, especially the existence of the conjugate function in this space. Finally, it shows an application of Jensen’s inequality to solve High School Olympic problems.