# Convex Analysis (AN11)

### General

School School of Science Department of Mathematics Graduate AN11 2 Convex Analysis Lectures (Weekly Teaching Hours: 3, Credits: 7.5) Specialized general knowledge Real Analysis, Calculus I and Calculus II Greek Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes The course aims to an introduction to convex analysis at graduate level. Material varies from classical topics on convex analysis to recent research problems. The students should: get knowledge on issues from a wide area of topics on convex analysis, aim ability to start research on problems on the theory of convex analysis, be introduced to the literature on problems in the area of convex analysis that he/she was taught. Working independently. Team work. Production of free, creative and inductive thinking. Production of analytic and synthetic thinking. Search for, analysis and synthesis of data and information, with the use of the necessary technology. Get in touch with specialized knowledge and evolve abilities for comparing, obtaining and evaluating results on the specific area of interest.

### Syllabus

Basic notions. Convex functions and convex sets, convexity criteria. Normed spaces. Dual spaces and the Legendre transform. The Caratheodory Theorem and its applications to geometry. Radon’s and Helly’s theorems. Minkowski’s First theorem and its applications to Optimization Theory. The concentration of measure phenomenon on the sphere. Dvoretzky’s theorem and the Quotient of Subspace theorem. The Brunn-Minkowski inequality and its generalizations (Lp variants and functional forms). Mixed volumes and inequalities of Aleksandrov-Fenchel type. Inequalities of isoperimetric type (e.g. the classical isoperimetric inequality and the Blaschke-Santalo inequality). The Brascamp-Lieb inequality and reverse isoperimetric inequalities. Area measures of convex hypersurfaces. The Minkowski Existence and Uniqueness problem and its generalizations, applications to the Theory of Monge-Ampere equations. Classical open problems.

### Teaching and Learning Methods - Evaluation

Delivery

Lectures/ Class presentations

Use of Information and Communications Technology -
Teaching Methods
Lectures/Presentations 45
Assignments/Essays 52.5
Individual study 90
Course total 187.5
Student Performance Evaluation

Students choose evaluation by one or both of the following:

• Class presentation - Essays - Assignments
• Final Written Examination

In case that a student participates to both, the final grade is the maximum of the two grades. Evaluation criteria and all steps of the evaluation procedure will be accessible to students through the platform “E-course” of the University of Ioannina.

### Attached Bibliography

• J. Bakelman, Convex Analysis And Nonlinear Geometric Elliptic Equations
• R. J. Gardner, Geometric tomography. Second edition.
• H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics.
• Koldobsky, Fourier Analysis in Convex Geometry.
• M. Ledoux, The Concentration of Measure Phenomenon.
• V.D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces
• R. Tyrel Rockafellar, Convex Analysis.
• R. Schneider, Convex bodies: the Brunn-Minkowski theory. Second expanded edition.
• R. Schneider and W. Weil, Stochastic and Integral Geometry.
• C. Thompson, Minkowski Geometry.