Convex Analysis (MAE753)

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑE817
Semester	8
Course Title	Convex Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	In the platform "E-course" of the University of Ioannina

Learning Outcomes

Learning outcomes	The course aims to an introduction to convex analysis at undergraduate level. It is desired for students to understand convex sets with respect to some of their qualitative (from a geometric/combinatorial point of view) and quantitative (e.g. volume, surface area) properties together with the study of the corresponding convex functions.
General Competences	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. Polytopes. Gauge functions and support functions. The Caratheodory. Radon's and Helly's theorems. Minkowski's First theorem. The Brunn-Minkowski inequality. Mixed volumes. Inequalities of isoperimetric type (e.g. the classical isoperimetric inequality and the Blaschke-Santalo inequality). F. John's Theorem. The reverse isoperimetric inequality.

Teaching and Learning Methods - Evaluation

Delivery	Lectures/ Class presentations
Use of Information and Communications Technology	Use of the platform “E-course” of the University of Ioannina
Teaching Methods	Activity Semester Workload Lectures/Presentations 39 Assignments/Essays 33 Individual study 78 Course total 150
Student Performance Evaluation	Students' evaluation by the following: Class presentation - Essays - Assignments Final Written Examination 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

• R. J. Gardner, Geometric tomography. Second edition.
• R. Tyrel Rockafellar, Convex Analysis.
• R. Schneider, Convex bodies: the Brunn-Minkowski theory. Second expanded edition.
• A. C. Thompson, Minkowski Geometry.
• R. Webster, Convexity.