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1	[en] ISOPERIMETRIC PROBLEMS IN THE MINKOWSKI PLANE / [pt] PROBLEMAS ISOPERIMÉTRICOS NO PLANO DE MINKOWSKI MARCELO CHAVES SILVA 13 January 2016 (has links) [pt] O objetivo principal deste trabalho é resolver o problema isoperimétrico no plano de Minkowski, isto é, determinar dentre todas as curvas convexas, fechadas, simples e suaves de perímetro fixo de um plano munido com uma norma qualquer, qual é aquela que delimita a maior área. Mostraremos que a solução para este problema não é necessariamente o círculo como no caso euclideano e sim uma curva conhecida como isoperimetrix. Para isto, vamos demonstrar a desigualdade de Minkowski a partir do conceito de área mista. Em seguida, vamos determinar se há outros casos (além do caso euclideano) em que o círculo coincide com o isoperimetrix. Também iremos mostrar que o perímetro da bola nestes planos pode assumir qualquer valor real entre seis e oito, sendo seis quando a bola for um hexágono regular afim e oito quando for um paralelogramo. / [en] The main objective of this work is to solve the isoperimetric problem in the Minkowski plane, i. e., determine among all smooth simple closed convex curves of a normed plane with fixed perimeter, what is that which defines the largest area. We will show that the solution to this problem is not necessarily the circle as in the Euclidean case, but a curve known as isoperimetrix. For this, we will demonstrate the Minkowski inequality from the concept of mixed area. Then, we determine if there are other cases (apart from the Euclidean case) in which the circle coincides with the isoperimetrix. We will also show that the ball perimeter in a normed plane can take any real value between six and eight. It is six when the ball is an affine regular hexagon and eight when it is a parallelogram. [pt] GEOMETRIA DIFERENCIAL [en] DIFFERENTIAL GEOMETRY [pt] PLANO NORMADO [en] NORMED PLANE [pt] NORMA DUAL [pt] DESIGUALDADE DE MINKOWSKI [en] MINKOWSKI INEQUALITY
2	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
3	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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