Selected problems from minkowski geometry [Elektronische Ressource] / vorgelegt von Nico Düvelmeyer

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_chemnitz
• Publié le: 01 janvier 2006
• Nombre de lectures: 17
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Selected Problems
from
Minkowski Geometry
von der Fakultat fur Mathematik¨ ¨
der Technischen Universitat Chemnitz genehmigte¨
DISSERTATION
zur Erlangung des akademischen Grades
Doctor rerum naturalium
(Dr. rer. nat.)
¨TECHNISCHE UNIVERSITAT CHEMNITZ
Fakultat fur Mathematik¨ ¨
vorgelegt von Dipl.-Math. Nico Duvelmeyer¨
geb. am 30. Mai 1979 in Greiz eingereicht am 8. Juni 2006
gefo¨rdert durch ein Stipendium der Studienstiftung des deutschen Volkes (4/2003 bis 3/2005)
Gutachter: Prof. Dr. H. Martini (Betreuer, TU Chemnitz)
Prof. Dr. K. J. Swanepoel (UNISA, Pretoria, South Africa)
Prof. Dr. E. Hertel (Friedrichchiller-Universit a¨t Jena)
Tag der Verteidigung: 9. November 2006
Verfug¨ bar im MONARCH der TU Chemnitz: http://archiv.tu-chemnitz.de/pub/2006/0196Preface
The results of this dissertation refer to the geometry of Minkowski spaces, i.e., of finite dimensional
normed linear spaces. Also in view of very recent developments, this field can be located at the
intersectionofFinslerGeometry,BanachSpaceTheory,andConvexGeometry,butitisalsoclosely
related to Distance Geometry and Abstract Convexity. Moreover, the main part of the obtained
results belongs to the Discrete Geometry of normed planes, where the use of numerical methods
is new in this field. Having chosen such an approach, we follow the modern trend that numerical
computations are based on exact arithmetics. Within these computations semi-algebraic subsets
of the real numbers occur frequently as basic mathematical objects.
More precisely, the results obtained here can be classified to belong to the following topics:
Discrete and Convex Geometry (MR 52), including the theory of polytopes, finite dimensional
Banach Space Theory (MR 46Bxx), and Foundations of (nonuclidean) Geometries (MR 51 and
MR 53).
Many specialists in the field of Discrete Geometry know the problem of classifying 2istance
sets. Wepresentsuchaclassification,whichcanbeconsideredasthemainresultofthedissertation.
Due to several talks of myself at international conferences, these experts are waiting with large
interest for this complete classification of 2istance sets. It occurs here for the first time in printed
form. Otherpartsofthedissertationrefertocharacterizationsofinnerproductspacesbygeometric
properties of the space and related subjects.
Acknowledgement
My work was guided by valuable questions and hints of Prof. Dr. Horst Martini and Prof. Dr.
Konrad J. Swanepoel. I am also grateful for helpful discussions with Gennadiy Averkov.
Finally, I want to thank the “Studienstiftung des deutschen Volkes” for financial support.
1Contents
Introduction 5
1 Prerequisites 6
1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Topological notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Algebraic notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.2 The Euclidean metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.3 Topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.4 Affine geometric notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.5 Linear geometric notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Algebraic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.8 Minkowski geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.8.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.9 Birkhoff orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.10 The functional β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.11 Parametrization of the unit circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.12 Tangent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.13 Radon curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.14 Equiframed bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.15 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.16 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.17 Polyhedra and polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.18 Tasks, systems, and algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Angular measures and bisectors 19
2.1 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Properties of angular bisectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2CONTENTS 3
2.4.1 Proofs concerning the properties of Busemann and Glogovskij angular bisector 22
2.4.2 Parametrization of the unit circle . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.3 The equivalence of Busemann and Glogovskij angular bisectors . . . . . . . 24
2.4.4 Extensions of systems of angular bisectors for straight angles . . . . . . . . 24
2.4.5 The equivalence of Busemann’s definition with that of a ?isector . . . . . 25
2.4.6 The equivalence of Glogovskij’s definition with that of a ?isectors . . . . 28
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Convex bodies with equiframed 2-dimensional sections 30
3.1 The statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Indirect approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Planar considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Local 3imensional extensions of the planar properties . . . . . . . . . . . . . . . 32
3.5 Global 3imensional properties of ∂B . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Interpretation of the contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.7 Application to the results for angular measures . . . . . . . . . . . . . . . . . . . . 36
4 Embedding metric spaces into a Minkowski space 37
4.1 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Transformation of the embedding task . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2.1 Embedding into a suitable Minkowski space . . . . . . . . . . . . . . . . . . 38
4.2.2 Embedding into a given polytopal Minkowski space . . . . . . . . . . . . . . 42
4.3 Simplifying the analytical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.1 General simplification principles . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.2 Simplifying the general embedding system . . . . . . . . . . . . . . . . . . . 45
4.3.3 Using subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Algorithmical solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Reducing the number of systems which need to be checked for admissibility . . . . 47
4.6 One application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.7 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 Solving parametrized linear systems 56
5.1 Systems to solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1.1 Homogeneous and inhomogeneous systems . . . . . . . . . . . . . . . . . . . 56
5.1.2 Removing strict inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.3 Coefficients of the linear functions . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.4 Polynomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.5 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Solution of a linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.1 Admissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.2 Solution vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.3 Solution set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58CONTENTS 4
5.2.4 Special cases for the influence of parameters . . . . . . . . . . . . . . . . . . 60
5.3 Certificates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3.1 Certificates for nondmissibility . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3.2 Certificates for upper bounds on the dimension of the solution set . . . . . 62
5.3.3 Certificates for lower bounds on the dimension of the solution set . . . . . . 63
5.3.4 Certificate for full solution