(in Polish) Analiza wypukła

mathematics

elective monographs

Functional Analysis 1000-135AF
Introduction to Differential Geometry 1000-135WGR

Good knowledge of basic multivariate real analysis and linear algebra as being taught in the first and second year of undergraduate studies.

Classroom

Convex functions constitute an interesting and rich class of objects. Their key properties can be summarized as follows: each local minimum is global and the maximum on a bounded set is assumed on the boundary. These features make them ideal, e.g., for investigating optimization problems. In the lecture we will deal with the classical (elementary) theory in finite dimensional Euclidean spaces as it is presented in the book "Convex analysis" by R. T. Rockafellar. Certain infinite dimensional Banach spaces may, however, appear in the exercises.

The main topics are:

* convex conjugation (Fenchel transform)

* Legendre transform

* polarity of convex cones

* The Alexandrov theorem on the existence of the second Taylor polynomial almost everywhere

* The Carathéodory representation theorem for convex envelopes

* min-max problems

* The Fenchel duality theorem

We will start with the characterization of convex functions and sets and define the basic operations on these objects. Next, we will discuss their topological properties. After that, we will focus on the duality between points and hypersurfaces, and introduce convex conjugation (Fenchel transform) and polarity operations. When examining convex cones, we will also focus for a moment on the relationship with the theory of norms. We will further prove The Carathéodory representation theorem for points in the convex hull of some set and also deal with the concept of extreme points. We shall then move on to the differentiability of convex functions, define the subgradient, discuss the Legendre transform, and examine its relationship to convex conjugation. We will examine when the gradient of a convex function defines the homeomorphism of its domain with the domain of the conjugate. Perhaps it will also be possible to prove The Alexandrov theorem about the existence of a second Taylor polynomial for a convex function in almost every point of its domain. Finally, if time permits, we will tackle the topic of convex minimization with convex constraints and the min-max problem for the convex-concave function. In particular, we will prove the Fenchel duality theorem.

In the exercises, apart from solving the tasks illustrating and supplementing the lecture, we will look at a certain generalization of convex sets, namely sets of positive reach.

Primary:

* "Convex analysis" R. T. Rockafellar

* "Fundatnentals of Convex Analysis" J-B. Hiriart-Urruty, C. Lemarechal

Secondary:

* "Minkowski Geometry" A. C. Thompson

* "Lectures on Convex Geometry" D. Hug, W. Weil

* "Variational Analysis" R. T. Rockafellar, R. J-B. Wets

* "Convex Functions and their Applications: A Contemporary Approach" C. P. Niculescu, L-E. Persson

* "Curvature measures" H. Federer

* "User’s guide to viscosity solutions of second order partial differential equations" M. Crandall, H. Ishii, P. Lions

* "Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets" A.D. Rosa, S. Kolasiński, M. Santilli

* Knowledge of basic tools and theorems for finite dimensional convex analysis.

* Awareness of the relationship between convex analysis and calculus of variations and optimization problems.

* Understanding the relationship between convex analysis, convex geometry, and the theory of normed spaces.

* Knowledge of some applications, e.g. for studying sets of positive reach.

To be admitted to the exam one needs to present, in exercise sessions, certain number of solutions of given problems (homework).

The final exam shall be oral but the list of questions / topics / problems will be given in advance (at the end of the semester).