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1	Periodische Verschachtelungsprobleme in x [0,w] / Garms, Onno. January 2006 (has links) Bonn, Universiẗat, Diss., 2006. Minkowski-Geometrie. Minkowski-Summe.
2	Selected Problems from Minkowski Geometry Düvelmeyer, Nico 24 November 2006 (has links) (PDF) Die Dissertation behandelt zwei Gebiete der Geometrie endlichdimensionaler Banach-Räume (Minkowski-Geometrie). Der erste Schwerpunkt liegt dabei auf Winkelmassen und Winkelhalbierenden. Dafür gibt es verschiedene Verallgemeinerungen dieser Euklidischen Konzepte, die im allgemeinen in Minkowski-Räumen verschieden sind. Es werden alle Minkowski-Räume charakterisiert, in welchen zwei dieser Konzepte für alle möglichen Winkel das selbe Maß oder die selben Winkelhalbierenden liefern. Der zweite Teil der Dissertation behandelt die Einbettung von metrischen Räumen in Minkowski-Räume. Dabei steht die Einbettung in beliebige geeignete Minkowski-Räume fester Dimension im Mittelpunkt. Hauptergebnis ist hier die vollständige Klassifikation aller 2-Abstands-Mengen in Minkowski-Ebenen, d.h., aller möglichen Mengen von Punkten einer Minkowski-Ebene, so dass zwischen diesen Punkten nur zwei verschiedene positive Abstandswerte auftreten. / This dissertation deals with two geometric subjects in finite dimensional Banach spaces (Minkowski geometry). The first topics are angle measures and angular bisectors. There are several possibilities to generalize these Euclidean concepts, which yield in general distinct geometrical objects in Minkowski spaces. A characterization is given for Minkowski spaces, for which two such concepts yield for all possible angles the same angular measure or the same angular bisector. The second part of the dissertation deals with embeddings of metric spaces into Minkowski spaces. It focuses on embeddings into some arbitrary suitable Minkowski space of prescribed dimension. The major result is the complete classification of all 2-distance sets in Minkowski planes, i.e., of all subsets of points of a Minkowski plane such that there are only two different positive distance values between these points. 2-Abstands-Menge ddc:510 Einbettungsproblem Minkowski-Geometrie Ungleichungssystem Winkel
3	Selected Problems from Minkowski Geometry Düvelmeyer, Nico 11 September 2006 (has links) Die Dissertation behandelt zwei Gebiete der Geometrie endlichdimensionaler Banach-Räume (Minkowski-Geometrie). Der erste Schwerpunkt liegt dabei auf Winkelmassen und Winkelhalbierenden. Dafür gibt es verschiedene Verallgemeinerungen dieser Euklidischen Konzepte, die im allgemeinen in Minkowski-Räumen verschieden sind. Es werden alle Minkowski-Räume charakterisiert, in welchen zwei dieser Konzepte für alle möglichen Winkel das selbe Maß oder die selben Winkelhalbierenden liefern. Der zweite Teil der Dissertation behandelt die Einbettung von metrischen Räumen in Minkowski-Räume. Dabei steht die Einbettung in beliebige geeignete Minkowski-Räume fester Dimension im Mittelpunkt. Hauptergebnis ist hier die vollständige Klassifikation aller 2-Abstands-Mengen in Minkowski-Ebenen, d.h., aller möglichen Mengen von Punkten einer Minkowski-Ebene, so dass zwischen diesen Punkten nur zwei verschiedene positive Abstandswerte auftreten. / This dissertation deals with two geometric subjects in finite dimensional Banach spaces (Minkowski geometry). The first topics are angle measures and angular bisectors. There are several possibilities to generalize these Euclidean concepts, which yield in general distinct geometrical objects in Minkowski spaces. A characterization is given for Minkowski spaces, for which two such concepts yield for all possible angles the same angular measure or the same angular bisector. The second part of the dissertation deals with embeddings of metric spaces into Minkowski spaces. It focuses on embeddings into some arbitrary suitable Minkowski space of prescribed dimension. The major result is the complete classification of all 2-distance sets in Minkowski planes, i.e., of all subsets of points of a Minkowski plane such that there are only two different positive distance values between these points. info:eu-repo/classification/ddc/510 ddc:510 Einbettungsproblem Minkowski-Geometrie Ungleichungssystem Winkel 2-Abstands-Menge
4	Discrete Geometry in Normed Spaces Spirova, Margarita 09 December 2010 (has links) (PDF) This work refers to ball-intersections bodies as well as covering, packing, and kissing problems related to balls and spheres in normed spaces. A quick introduction to these topics and an overview of our results is given in Section 1.1 of Chapter 1. The needed background knowledge is collected in Section 1.2, also in Chapter 1. In Chapter 2 we define ball-intersection bodies and investigate special classes of them: ball-hulls, ball-intersections, equilateral ball-polyhedra, complete bodies and bodies of constant width. Thus, relations between the ball-hull and the ball-intersection of a set are given. We extend a minimal property of a special class of equilateral ball-polyhedra, known as Theorem of Chakerian, to all normed planes. In order to investigate bodies of constant width, we develop a concept of affine orthogonality, which is new even for the Euclidean subcase. In Chapter 2 we solve kissing, covering, and packing problems. For a given family of circles and lines we find at least one, but for some families even all circles kissing all the members of this family. For that reason we prove that a strictly convex, smooth normed plane is a topological Möbius plane. We give an exact geometric description of the maximal radius of all homothets of the unit disc that can be covered by 3 or 4 translates of it. Also we investigate configurations related to such coverings, namely a regular 4-covering and a Miquelian configuration of circles. We find the concealment number for a packing of translates of the unit ball. Packungen convex bodies bodies of constant width coverings packings Minkowski geometry normed spaces ddc:510 konvexer Körper Körper konstanter Breite Überdeckung Minkowski-Geometrie normierter Raum Packung <Mathematik>
5	Discrete Geometry in Normed Spaces Spirova, Margarita 02 December 2010 (has links) This work refers to ball-intersections bodies as well as covering, packing, and kissing problems related to balls and spheres in normed spaces. A quick introduction to these topics and an overview of our results is given in Section 1.1 of Chapter 1. The needed background knowledge is collected in Section 1.2, also in Chapter 1. In Chapter 2 we define ball-intersection bodies and investigate special classes of them: ball-hulls, ball-intersections, equilateral ball-polyhedra, complete bodies and bodies of constant width. Thus, relations between the ball-hull and the ball-intersection of a set are given. We extend a minimal property of a special class of equilateral ball-polyhedra, known as Theorem of Chakerian, to all normed planes. In order to investigate bodies of constant width, we develop a concept of affine orthogonality, which is new even for the Euclidean subcase. In Chapter 2 we solve kissing, covering, and packing problems. For a given family of circles and lines we find at least one, but for some families even all circles kissing all the members of this family. For that reason we prove that a strictly convex, smooth normed plane is a topological Möbius plane. We give an exact geometric description of the maximal radius of all homothets of the unit disc that can be covered by 3 or 4 translates of it. Also we investigate configurations related to such coverings, namely a regular 4-covering and a Miquelian configuration of circles. We find the concealment number for a packing of translates of the unit ball. info:eu-repo/classification/ddc/510 ddc:510 Packungen
6	Metrical Problems in Minkowski Geometry Fankhänel, Andreas 19 October 2012 (has links) (PDF) In this dissertation we study basic metrical properties of 2-dimensional normed linear spaces, so-called (Minkowski or) normed planes. In the first chapter we introduce a notion of angular measure, and we investigate under what conditions certain angular measures in a Minkowski plane exist. We show that only the Euclidean angular measure has the property that in an isosceles triangle the base angles are of equal size. However, angular measures with the property that the angle between orthogonal vectors has a value of pi/2, i.e, a quarter of the full circle, exist in a wider variety of normed planes, depending on the type of orthogonality. Due to this we have a closer look at isosceles and Birkhoff orthogonality. Finally, we present results concerning angular bisectors. In the second chapter we pay attention to convex quadrilaterals. We give definitions of different types of rectangles and rhombi and analyse under what conditions they coincide. Combinations of defining properties of rectangles and rhombi will yield squares, and we will see that any two types of squares are equal if and only if the plane is Euclidean. Additionally, we define a ``new\'\' type of quadrilaterals, the so-called codises. Since codises and rectangles coincide in Radon planes, we will explain why it makes sense to distinguish these two notions. For this purpose we introduce the concept of associated parallelograms. Finally we will deal with metrically defined conics, i.e., with analogues of conic sections in normed planes. We define metric ellipses (hyperbolas) as loci of points that have constant sum (difference) of distances to two given points, the so-called foci. Also we define metric parabolas as loci of points whose distance to a given point equals the distance to a fixed line. We present connections between the shape of the unit ball B and the shape of conics. More precisely, we will see that straight segments and corner points of B cause, under certain conditions, that conics have straight segments and corner points, too. Afterwards we consider intersecting ellipses and hyperbolas with identical foci. We prove that in special Minkowski planes, namely in the subfamily of polygonal planes, confocal ellipses and hyperbolas intersect in a way called Birkhoff orthogonal, whenever the respective ellipse is large enough. Winkelmaß Birkhoff-Orthogonalität Codis Kegelschnitt gleichschenklig-rechtwinklig metrische Ellipse metrische Hyperbel metrische Parabel metrisches Parallelogramm Minkowski-Ebene normierte Ebene Parallelogramm Radon-Ebene Rechteck Rhombus glatte Ebene Quadrat streng konvexe Ebene angular measure Birkhoff orthogonality codis conic section isosceles orthogonality metric ellipse metric hyperbola metric parabola metric parallelogram Minkowski plane normed plane parallelogram Radon plane rectangle rhombus smooth plane square strictly convex plane ddc:516 Minkowski-Geometrie Kegelschnitt Viereck Winkel
7	Metrical Problems in Minkowski Geometry Fankhänel, Andreas 07 June 2012 (has links) In this dissertation we study basic metrical properties of 2-dimensional normed linear spaces, so-called (Minkowski or) normed planes. In the first chapter we introduce a notion of angular measure, and we investigate under what conditions certain angular measures in a Minkowski plane exist. We show that only the Euclidean angular measure has the property that in an isosceles triangle the base angles are of equal size. However, angular measures with the property that the angle between orthogonal vectors has a value of pi/2, i.e, a quarter of the full circle, exist in a wider variety of normed planes, depending on the type of orthogonality. Due to this we have a closer look at isosceles and Birkhoff orthogonality. Finally, we present results concerning angular bisectors. In the second chapter we pay attention to convex quadrilaterals. We give definitions of different types of rectangles and rhombi and analyse under what conditions they coincide. Combinations of defining properties of rectangles and rhombi will yield squares, and we will see that any two types of squares are equal if and only if the plane is Euclidean. Additionally, we define a ``new\'\' type of quadrilaterals, the so-called codises. Since codises and rectangles coincide in Radon planes, we will explain why it makes sense to distinguish these two notions. For this purpose we introduce the concept of associated parallelograms. Finally we will deal with metrically defined conics, i.e., with analogues of conic sections in normed planes. We define metric ellipses (hyperbolas) as loci of points that have constant sum (difference) of distances to two given points, the so-called foci. Also we define metric parabolas as loci of points whose distance to a given point equals the distance to a fixed line. We present connections between the shape of the unit ball B and the shape of conics. More precisely, we will see that straight segments and corner points of B cause, under certain conditions, that conics have straight segments and corner points, too. Afterwards we consider intersecting ellipses and hyperbolas with identical foci. We prove that in special Minkowski planes, namely in the subfamily of polygonal planes, confocal ellipses and hyperbolas intersect in a way called Birkhoff orthogonal, whenever the respective ellipse is large enough.:1 Introduction 2 On angular measures 3 Types of convex quadrilaterals 4 On conic sections info:eu-repo/classification/ddc/516 ddc:516

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