11 |
Discrete Geometry in Normed SpacesSpirova, 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.
|
12 |
Geometry of Minkowski Planes and Spaces -- Selected TopicsWu, Senlin 13 November 2008 (has links)
The results presented in this dissertation refer to the geometry of Minkowski
spaces, i.e., of real finite-dimensional Banach spaces.
First we study geometric properties of radial projections of
bisectors in Minkowski spaces, especially the relation between the
geometric structure of radial projections and Birkhoff
orthogonality. As an application of our results it is shown that for
any Minkowski space there exists a number, which plays somehow the
role that $\sqrt2$ plays in Euclidean space. This number is referred
to as the critical number of any Minkowski space. Lower and upper
bounds on the critical number are given, and the cases when these
bounds are attained are characterized. Moreover, with the help of
the properties of bisectors we show that a linear map from a normed
linear space $X$ to another normed linear space $Y$ preserves
isosceles orthogonality if and only if it is a scalar multiple of a
linear isometry.
Further on, we examine the two tangent segments from any exterior
point to the unit circle, the relation between the length of a chord
of the unit circle and the length of the arc corresponding to it,
the distances from the normalization of the sum of two unit vectors
to those two vectors, and the extension of the notions of
orthocentric systems and orthocenters in Euclidean plane into
Minkowski spaces. Also we prove theorems referring to chords of
Minkowski circles and balls which are either concurrent or parallel.
All these discussions yield many interesting characterizations of
the Euclidean spaces among all (strictly convex) Minkowski spaces.
In the final chapter we investigate the relation between the length
of a closed curve and the length of its midpoint curve as well as
the length of its image under the so-called halving pair
transformation. We show that the image curve under the halving pair
transformation is convex provided the original curve is convex.
Moreover, we obtain several inequalities to show the relation
between the halving distance and other quantities well known in
convex geometry. It is known that the lower bound for the geometric
dilation of rectifiable simple closed curves in the Euclidean plane
is $\pi/2$, which can be attained only by circles. We extend this
result to Minkowski planes by proving that the lower bound for the
geometric dilation of rectifiable simple closed curves in a
Minkowski plane $X$ is analogously a quarter of the circumference of
the unit circle $S_X$ of $X$, but can also be attained by curves
that are not Minkowskian circles. In addition we show that the lower
bound is attained only by Minkowskian circles if the respective norm
is strictly convex. Also we give a sufficient condition for the
geometric dilation of a closed convex curve to be larger than a
quarter of the perimeter of the unit circle.
|
13 |
Discrete Geometry in Normed SpacesSpirova, 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.
|
Page generated in 0.0737 seconds