An Isoperimetric Theorem in Plane Geometry

   Abstract. Let P be a simple polygon. Let the vertices of P be mapped, according to a counterclockwise traversal of the boundary, into a strictly increasing sequence of real numbers in [0, 2π) . Let a ray be drawn from each vertex so that the angle formed by the ray and a horizontal line pointing to the right equals, in measure, the number mapped to the vertex. Whenever the rays from two consecutive vertices intersect, let them induce the triangular region with extreme points comprising the vertices and the intersection point. It is shown that there is a fixed α such that if all of the assigned angles are increased by α , the triangular regions induced by the redirected rays cover the interior of P . This covering implies the standard isoperimetric inequalities in two dimensions, as well as several new inequalities, and resolves a question posed by Yaglom and Boltanskii.     

A Degenerate Isoperimetric Problem in the Plane

We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and examples are provided. This continues the investigation begun in Alama et al. (Commun Pure Appl Math 70:340–377, 2017) where the metric \(ds^2\) near the singularities equals a quadratic polynomial times the standard metric. Here, we allow the conformal factor to be any smooth non-negative potential vanishing at isolated points provided the Hessian at these points is positive definite. These isoperimetric curves, appropriately parametrized, arise as traveling wave solutions to a bi-stable Hamiltonian system.     

Isoperimetric Constants of Infinite Plane Graphs

   Abstract. Let G be an infinite locally finite plane graph with one end and let H be a finite plane subgraph of G . Denote by a(H) the number of finite faces of H and by l(H) the number of the edges of H that are on the boundary of the infinite face or a finite face not in H . Define the isoperimetric constant h (G) to be inf _H l(H) / a(H) and define the isoperimetric constant h (δ) to be inf _G h (G) where the infimum is taken over all infinite locally finite plane graphs G having minimum degree δ and exactly one end. We establish the following bounds on h (δ) for δ ≥ 7 :

On the Dido Problem and Plane Isoperimetric Problems

This paper is a continuation of a series of papers, dealing with contact sub-Riemannian metrics on R³. We study the special case of contact metrics that correspond to isoperimetric problems on the plane. The purpose is to understand the nature of the corresponding optimal synthesis, at least locally. It is equivalent to studying the associated sub-Riemannian spheres of small radius. It appears that the case of generic isoperimetric problems falls down in the category of generic sub-Riemannian metrics that we studied in our previous papers (although, there is a certain symmetry). Thanks to the classification of spheres, conjugate-loci and cut-loci, done in those papers, we conclude immediately. On the contrary, for the Dido problem on a 2-d Riemannian manifold (i.e. the problem of minimizing length, for a prescribed area), these results do not apply. Therefore, we study in details this special case, for which we solve the problem generically (again, for generic cases, we compute the conjugate loci, cut loci, and the shape of small sub-Riemannian spheres, with their singularities). In an addendum, we say a few words about: (1) the singularities that can appear in general for the Dido problem, and (2) the motion of particles in a nonvanishing constant magnetic field.     

An Asymptotic Isoperimetric Inequality

For a finite metric space V with a metric , let V ⁿ be the metric space in which the distance between (a ₁ , . . ., a _n ) and (b ₁ , . . ., b _n ) is the sum . We obtain an asymptotic formula for the logarithm of the maximum possible number of points in V ⁿ of distance at least d from a set of half the points of V ⁿ , when n tends to infinity and d satisfies . Submitted: September 1997, Final version: November 1997     

Geometry:The Coordinate Plane

<正>When am I ever going to use this?MAPS A map of Terrell's neighborhood is shown. 1.Suppose Terrell starts at the corner of Russel and Main and walks 1block north and blocks east.Name the intersection of his location.     

Duality for Curves in Affine Plane Geometry

The dual of a plane curve in equiaffine geometry is defined as a curve in the space of conics. The image and inverse of this duality are described. It is shown that the duality transforms equiaffine vertices into cusps. As an application, an analogue of Kneser's lemma for osculating conics is proved.     

Constructive Axiomatization of Plane Hyperbolic Geometry

We provide a universal axiom system for plane hyperbolic geometry in a firstorder language with two sorts of individual variables, ‘points’ (upper‐case) and ‘lines’ (lowercase), containing three individual constants, A₀, A₁, A₂, standing for three non‐collinear points, two binary operation symbols, φ and ι, with φ(A, B) = l to be interpreted as ‘𝓁 is the line joining A and B’ (provided that A ≠ B, an arbitrary line, otherwise), and ι(g, h) = P to be interpreted as 𝓁P is the point of intersection of g and h (provided that g and h are distinct and have a point of intersection, an arbitrary point, otherwise), and two binary operation symbols, π₁(P, 𝓁) and —₂(P, 𝓁), with π_i(P, 𝓁) = g (for i = 1, 2) to be interpreted as ‘g is one of the two limiting paralle lines from P to 𝓁 (provided that P is not on 𝓁, an arbitrary line, otherwise).     

Conformal Geometry of Gravitational Plane Waves

By Brinkmann’s theorem the only Ricci flat and nonflat 4-manifolds admitting non-homothetic conformal vector fields are certain pp-waves. It seems that the converse direction was never completely clarified Which metrics really do occur? It is well known that the conformal group of a nonconformally-flat spacetime is atmost seven-dimensional and that seven is attained for certain pp-waves. Here we explicitly determine all solutions with a seven-dimensional conformal group. In other words We determine all Ricci flat Lorentzian manifolds admitting a seven-dimensional conformal group. They come in three particular families of gravitational plane waves. All of them are exact analytic solutions in terms of elementary functions. Furthermore, it turns out that Ricci flat Lorentzian manifolds with a six-dimensional conformal group are not necessarily real analytic.     

Curvature and Geometry of Tessellating Plane Graphs

We show that the growth of plane tessellations and their edge graphs may be controlled from below by upper bounds for the combinatorial curvature. Under the assumption that every geodesic path may be extended to infinity we provide explicit estimates of the growth rate and isoperimetric constant of distance balls in negatively curved tessellations. We show that the assumption about geodesics holds for all tessellations with at least p faces meeting in each vertex and at least q edges bounding each face, where (p,q) ∈ { (3,6), (4,4), (6,3) } . Received September 27, 1999, and in revised form May 3, 2000. Online publication September 22, 2000.     

A Resistance Bound Via An Isoperimetric Inequality

An isoperimetric upper bound on the resistance is given. As a corollary we resolve two problems, regarding mean commute time on finite graphs and resistance on percolation clusters. Further conjectures are presented.     

Enumerative Geometry for Plane Cubic Curves in Characteristic 2

Consider the plane cubic curves over an algebraically closed field of characteristic 2. By blowing up the parameter space P⁹ twice we obtain a variety B of complete cubics. We then compute the characteristic numbers for various families of cubics by intersecting cycles on B.     

关于平面卵形区域的等周亏格上界的几点注记

利用平面卵形区域的Ros'定理及其加强形式,给出平面R²中卵形区域的等周亏格的几个上界估计.     

从几何的角度看微分中值定理

从平面几何中曲线之间的相切关系不依赖于坐标轴的选取这一基本事实去看数学分析中Rolle中值定理、Lagrange中值定理以及Cauchy中值定理.