Concepts of curvatures in normed planes

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. To complete and clarify the whole picture, we show which known concepts are equivalent, and add also a new type of curvature. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behavior of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves.     

Constant Minkowskian width in terms of double normals

We extend the notion of a double normal of a convex body from smooth, strictly convex Minkowski planes to arbitrary two-dimensional real, normed, linear spaces in two different ways. Then, for both of these ways, we obtain the following characterization theorem: a convex body K in a Minkowski plane is of constant Minkowskian width iff every chord I of K splits K into two compact convex sets K₁ and K₂ such that I is a Minkowskian double normal of K₁ or K₂. Furthermore, the Euclidean version of this theorem yields a new characterization of d-dimensional Euclidean ball where d 3.     

A discrete four vertex theorem for hyperbolic polygons

There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a simple smooth closed curve in the plane has at least four extreme values. This theorem admits a natural discretization to Euclidean polygons due to O. Musin. In this article we adapt the techniques of Musin and prove a discrete four vertex theorem for convex hyperbolic polygons.     

On Reduced Triangles in Normed Planes

A convex body is reduced if it does not properly contain a convex body of the same minimal width. In this paper we present new results on reduced triangles in normed (or Minkowski) planes, clearly showing how basic seemingly elementary notions from Euclidean geometry (like that of the regular triangle) spread when we extend them to arbitrary normed planes. Via the concept of anti-norms, we study the rich geometry of reduced triangles for arbitrary norms giving bounds on their side-lengths and on their vertex norms. We derive results on the existence and uniqueness of reduced triangles, and also we obtain characterizations of the Euclidean norm by means of reduced triangles. In the introductory part we discuss different topics from Banach Space Theory, Discrete Geometry, and Location Science which, unexpectedly, benefit from results on reduced triangles.     

Planar acyclic oriented graphs

A plane Hasse representation of an acyclic oriented graph is a drawing of the graph in the Euclidean plane such that all arcs are straight-line segments directed upwards and such that no two arcs cross. We characterize completely those oriented graphs which have a plane Hasse representation such that all faces are bounded by convex polygons. From this we derive the Hasse representation analogue, due to Kelly and Rival of Fary's theorem on straight-line representations of planar graphs and the Kuratowski type theorem of Platt for acyclic oriented graphs with only one source and one sink. Finally, we describe completely those acyclic oriented graphs which have a vertex dominating all other vertices and which have no plane Hasse representation, a problem posed by Trotter.     

On the circular hull property in normed planes

We extend the notion of circular hull to arbitrary normed planes and prove that a compact, convex set of constant Minkowskian width has the circular hull property in such a plane. Also we show how this property is related to the so called weak circular intersection property.     

Circle geometry in affine Cayley-Klein planes

Some theorems from inversive and Euclidean circle geometry are extended to all affine Cayley-Klein planes. In particular, we obtain an analogue to the first step of Clifford’s chain of theorems, a statement related to Napoleon’s theorem, extensions of Wood’s theorem on similar-perspective triangles and of the known fact that the three radical axes of three given circles are parallel or have a point in common. For proving these statements, we use generalized complex numbers. Supported by a grant D01-761/24.10.06 from the Ministry of Education and Sciences, and by a grant 108/2007 from Sofia University.     

Projections,Birkhoff Orthogonality and Angles in Normed Spaces

Let $X$ be a Minkowski plane, i.e., a real two dimensional normed linear space. We use projections to give a definition of the angle $A_q(x, y)$ between two vectors $x$ and $y$ in $X$, such that $x$ is Birkhoff orthogonal to $y$ if and only if $A_q(x, y) =\frac{π}{2}$. Some other properties of this angle are also discussed.     

On Miquel’s theorem and inversions in normed planes

Asplund and Grünbaum proved that Miquel’s six-circles theorem holds in strictly convex, smooth normed planes if the considered circles have equal radii. We extend this result in two directions. First we prove that Miquel’s theorem for circles of equal radii (more precisely, a generalized version of it) is true in every normed plane, without the assumptions of strict convexity and smoothness, and give also some properties of the circle configuration related to this theorem. Second we clarify the situation if the circles of the corresponding configuration do not necessarily have equal radii.     

L0-线性函数的Hahn-Banach扩张定理的几何形式与随机赋范模中的Goldstine-Weston定理

本文给出了关于L⁰- 线性函数的Hahn-Banach 扩张定理的几何形式并证明这个几何形式等价于它的代数形式. 进一步, 我们利用这个几何形式给出了随机局部凸模中熟知的基本分离定理的一个新的且简单的证明. 最后, 利用这个分离定理, 我们同时在两种拓扑 —(ε, λ)- 拓扑和局部L⁰- 凸拓扑下证明了随机赋范模中的Goldstine-Weston 稠密性定理, 并举出一个反例说明在局部L⁰- 凸拓扑下如果随机赋范模不具有可数连接性质, 则Goldstine-Weston 稠密性定理不一定成立.     

A new type of orthogonality for normed planes

In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions d ⩾ 3.     

Singularities of Solution Surfaces for Quasilinear First-Order Partial Differential Equations

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.     

Zone diagrams in compact subsets of uniformly convex normed spaces

A zone diagram is a relatively new concept which has emerged in computational geometry and is related to Voronoi diagrams. Formally, it is a fixed point of a certain mapping, and neither its uniqueness nor its existence are obvious in advance. It has been studied by several authors, starting with T. Asano, J. Matoušek and T. Tokuyama, who considered the Euclidean plane with singleton sites, and proved the existence and uniqueness of zone diagrams there. In the present paper we prove the existence of zone diagrams with respect to finitely many pairwise disjoint compact sites contained in a compact and convex subset of a uniformly convex normed space, provided that either the sites or the convex subset satisfy a certain mild condition. The proof is based on the Schauder fixed point theorem, the Curtis-Schori theorem regarding the Hilbert cube, and on recent results concerning the characterization of Voronoi cells as a collection of line segments and their geometric stability with respect to small changes of the corresponding sites. Along the way we obtain the continuity of the Dom mapping as well as interesting and apparently new properties of Voronoi cells.     

An algorithm for constructing star-shaped drawings of plane graphs

A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing.In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.     

A perimeter-based angle measure in Minkowski planes

Measuring angles in the Euclidean plane is a well-known topic, but for general normed planes there exists a variety of different concepts. These can be of a special kind, e.g. also preserving special orthogonality types. But these concepts are no angle measures in the sense of measure theory since they are not additive. This motivates us to define a new angle measure for normed planes that is in fact a measure in the sense of measure theory. Furthermore, we look at related types of rotation and reflection.     

On the relative distances of seven points in a plane convex body

Let C be a convex body in the Euclidean plane. The relative distance of points p and q is twice the Euclidean distance of p and q divided by the Euclidean length of a longest chord in C with the direction, say, from p to q. We prove that, among any seven points of a plane convex body, there are two points at relative distance at most one, and one cannot be replaced by a smaller value. We apply our result to determine the diameter of point sets in normed planes. Zsolt Lángi: Partially supported by the Hung. Nat. Sci. Found. (OTKA), grant no. T043556 and T037752 and by the Alberta Ingenuity Fund.     

抛物线的一些性质

近年,在研究射影几何在二次曲线上的运用中,发现有些平面几何问题用射影几何研究更自然、条理更清楚,而用平面几何方法处理则有难度.将二次曲线中的抛物线放在拓广平面上,借助射影几何中的Pascal定理、Steiner定理,给出了抛物线一些有趣的性质.     

Halving circular arcs in normed planes

Studying the relation between the length of a chord of the unit circle and the length of the arc corresponding to it, some new characterizations of the Euclidean plane among all normed planes are obtained. All these results yield characterizations of inner product spaces in higher dimensions. Supported by the National Natural Science Foundation of China, grant number 10671048.