Ugrás a tartalomhoz

Convex Geometry

Csaba Vincze (2013)

University of Debrecen

Convex Geometry

Convex Geometry

Csaba Vincze

University of Debrecen, Hungary

TÁMOP-4.1.2.A/1-11/1 MSc Tananyagfejlesztés

Interdiszciplináris és komplex megközelítésű digitális tananyagfejlesztés a természettudományi képzési terület mesterszakjaihoz

2013


Tartalom

Előszó
Introduction
1. Elements
1.1 Linear Algebra
1.2 Topology
1.3 Affine and convex sets
1.4 Operations with sets
1.5 The Hausdorff distance
1.6 Convex functions
1.7 Excercises
2. Carathéodory's theorem
2.1 Affinely dependence and independence
2.2 Carathéodory's theorem
2.3 The colorful Carathéodory's theorem
2.4 Excercises
3. Helly's theorem
3.1 Radon's lemma
3.2 Tverberg's theorem
3.3 Helly's theorem
3.4 Excercises
4. Generalizations and Applications
4.1 Helly's theorem: generalizations and applications
4.2 Universal covers and approximately central symmetry
4.2.1 The Brunn-Minkowski inequality
4.3 A sandwich theorem
4.4 Excercises
5. Krasnosselsky's art gallery theorem
5.1 Krasnosselsky's art gallery theorem
5.2 Excercises
6. Intersections of star-shaped sets
6.1 Intersections of star-shaped sets
6.2 Excercises
7. Separating and supporting hyperplanes
7.1 Separating and supporting hyperplanes
7.2 Krein-Milman's theorem
7.3 Support functions and Minkowski functionals
7.3.1 Minkowski functionals
7.3.2 Support functions
7.3.3 Radon planes
7.4 Excercises
8. Kirchberger's separation theorem
8.1 Kirchberger's separation theorem
8.2 Separation by spherical surfaces
8.3 The best affine approximation
8.4 Excercises
9. Convex polytopes and polyhedra
9.1 Vertices, edges, faces
9.2 Euler's and Descartes theorem
9.3 Regular polyhedra
10. Generalized conics
10.1 A panoramic view
10.2 Special types of generalized conics
10.2.1 The case of reducible subgroups
10.2.2 The case of irreducible subgroups
10.3 Applications in Geometric Tomography
11. Erdős-Vincze's theorem
11.1 Polyellipses in the plane
11.2 On the curvature of polyellipses
11.3 Erdős-Vincze's theorem
12. Rådström's embedding theorem
12.1 Rådström's embedding theorem
13. Acknowledgement
14. References