Convex Analysis (AN11)
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 Department of Mathematics
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General
School  School of Science 

Academic Unit  Department of Mathematics 
Level of Studies  Graduate 
Course Code  AN11 
Semester  2 
Course Title  Convex Analysis 
Independent Teaching Activities  Lectures (Weekly Teaching Hours: 3, Credits: 7.5) 
Course Type  Specialized general knowledge 
Prerequisite Courses 
Real Analysis, Calculus I and Calculus II 
Language of Instruction and Examinations 
Greek 
Is the Course Offered to Erasmus Students  Yes (in English) 
Course Website (URL)  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:


General Competences 

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 BrunnMinkowski inequality and its generalizations (Lp variants and functional forms). Mixed volumes and inequalities of AleksandrovFenchel type. Inequalities of isoperimetric type (e.g. the classical isoperimetric inequality and the BlaschkeSantalo inequality). The BrascampLieb inequality and reverse isoperimetric inequalities. Area measures of convex hypersurfaces. The Minkowski Existence and Uniqueness problem and its generalizations, applications to the Theory of MongeAmpere equations. Classical open problems.
Teaching and Learning Methods  Evaluation
Delivery 
Lectures/ Class presentations  

Use of Information and Communications Technology    
Teaching Methods 
 
Student Performance Evaluation 
Students choose evaluation by one or both of the following:
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 “Ecourse” 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 BrunnMinkowski theory. Second expanded edition.
 R. Schneider and W. Weil, Stochastic and Integral Geometry.
 C. Thompson, Minkowski Geometry.