Affine and convex sets. Topological properties of convex sets. Theorems of Caratheodory and Radon. Separation theorems. Extreme points. Polytopes. Cones and polars. Polyhedra. The theorem of Weyl and Minkowski. Systems of linear inequations. The Farkas lemma and linear programming. Generalizations of the theorem of Helly. The metric space of convex sets. The Blaschke theorem. Metric properties of convex sets. Convex functions. Continuity, differentiability and the subgradient. Extrema.

# Convexity

H. G. Eggleston: Convexity, Cambridge Univ. Press, Cambridge, 1958.

A. Brøndsted: An Introduction to Convex Polytopes, Springer, New York-Berlin, 1983.

F. A. Valentine: Convex Sets, Robert E. Krieger, Huntington, 1976.

R. T. Rockafellar: Convex Analysis, Princeton Univ. Press, Princeton, 1996.

A. W. Roberts, D. E. Varberg: Convex Functions, Academic Press, New York-London, 1973.

The student learns the basic concepts of convex geometry and convex analysis. The student gets familiar with the properties of convex sets and convex functions in euclidean and normed spaces and applications of the theory in different areas of mathematics. The student combines geometric intuition with algebra, analysis and combinatorics.

Knowledge and understanding:

Knowledge and understanding of basic concepts of the theory of convex sets and convex functions. A synthesis of methods of linear algebra, analysis and geometry.

Application:

Solving problems in different areas of mathematics and other sciences using the theory.

Reflection:

Understanding the theory on the basis of examples and applications.

Transferable skills:

Posing of a problem, its mathematical formulation, solving and analysis. The transfer of the theory into praxis.

Lectures, exercises, homeworks, consultations

Type (examination, oral, coursework, project):

2 midterm exams instead of written exam, written exam

oral exam

Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Franc Forstnerič:

FORSTNERIČ, Franc. Runge approximation on convex sets implies the Oka property. Annals of mathematics, ISSN 0003-486X, 2006, vol. 163, no. 2, str. 689-707. [COBISS-SI-ID 13908825]

FORSTNERIČ, Franc. Noncritical holomorphic functions on Stein manifolds. Acta mathematica, ISSN 0001-5962, 2003, vol. 191, no. 2, str. 143-189. [COBISS-SI-ID 13138009]

FORSTNERIČ, Franc. Embedding strictly pseudoconvex domains into balls. Transactions of the American Mathematical Society, ISSN 0002-9947, 1986, let. 295, št. 1, str. 347-368. [COBISS-SI-ID 8206425]

Klemen Šivic:

KLEP, Igor, MCCULLOUGH, Scott, ŠIVIC, Klemen, ZALAR, Aljaž. There are many more positive maps than completely positive maps. International mathematics research notices. June 2019, vol. 2019, iss. 11, str. 3313-3375. ISSN 1073-7928. [COBISS-SI-ID 18670425]

KANDIĆ, Marko, ŠIVIC, Klemen. On the dimension of the algebra generated by two positive semi-commuting matrices. Linear Algebra and its Applications. [Print ed.]. 2017, vol. 512, str. 136-161. ISSN 0024-3795. [COBISS-SI-ID 17776985]

KUZMA, Bojan, OMLADIČ, Matjaž, ŠIVIC, Klemen, TEICHMANN, Josef. Exotic one-parameter semigroups of endomorphisms of a symmetric cone. Linear Algebra and its Applications. [Print ed.]. 2015, vol. 477, str. 42-75. ISSN 0024-3795. [COBISS-SI-ID 17257561]