Some Bonnesen-style inequalities for higher dimensions

We discuss the higher dimensional Bonnesen-style inequalities.Though there are many Bonnesen-style inequalities for domains in the Euclidean plane R2 few results for general domain in R n(n ≥ 3) are known.The results obtained in this paper are for general domains,convex or non-convex,in Rn.     

Integral representations of quermassintegrals and Bonnesen-style inequalities

Archiv der Mathematik -     

The role of the kernel in Bonnesen-style inradius inequalities

The aim of this note is to present a topological method for detecting symbolic dynamics based on the discrete Conley index.     

On the isoperimetric inequalities for Reuleaux polygons

We give a unifying approach to the Blaschke-Lebesgue Theorem and the Firey-Sallee Theorem on Reuleaux polygons in the Euclidean plane.     

Problems on polygons and Bonnesen-type inequalities

In this paper we are interested in some Bonnesen-type isoperimetric inequalities for plane n-gons in relation with the two conjectures proposed by P. Levy and X.M. Zhang.     

The Poincaré metric and isoperimetric inequalities for hyperbolic polygons

We prove several isoperimetric inequalities for the conformal radius (or equivalently for the Poincaré density) of polygons on the hyperbolic plane. Our results include, as limit cases, the isoperimetric inequality for the conformal radius of Euclidean -gons conjectured by G. Pólya and G. Szegö in 1951 and a similar inequality for the hyperbolic -gons of the maximal hyperbolic area conjectured by J. Hersch. Both conjectures have been proved in previous papers by the third author.

Our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by A. Yu. Solynin. We also employ the dissymmetrization transformation of V. N. Dubinin. As an important part of our proofs, we obtain monotonicity and convexity results for special combinations of the Euler gamma functions, which appear to have a significant interest in their own right.

     

Wrapping polygons in polygons

This paper is developed toI ₂(2g).c-geometries, namely, point-line-plane structures where planes are generalized 2g-gons with exactly two lines on every point and any two intersecting lines belong to a unique plane.I ₂(2g).c-geometries appear in several contexts, sometimes in connection with sporadic simple groups. Many of them are homomorphic images of truncations of geometries belonging to Coxeter diagrams. TheI ₂(2g).c-geometries obtained in this way may be regarded as the standard ones. We characterize them in this paper. For everyI ₂(2g).c-geometry , we define a numberw(), which counts the number of times we need to walk around a 2g-gon contained in a plane of , building up a wall of planes around it, before closing the wall. We prove thatw()=1 if and only if is standard and we apply that result to a number of special cases.     

Bonnesen-style Isop erimetric Inequalities of an n-simplex

In this paper, by the theory of geometric inequalities, some new Bonnesen-style isoperimetric inequalities of n-dimensional simplex are proved. In several cases, these inequalities imply characterizations of regular simplex.     

Point retrieval for polygons

We present a solution to the following polygon retrieval problem: given a set of n points on the plane, build a data structure so that for any query polygon P the set of points lying in P can be retrieved efficiently. Our solution uses space O(n²) and reports the s points lying in a k-sided polygon P in time O(k log n + s).     

Extremal problems for convex polygons

Consider a convex polygon V _n with n sides, perimeter P _n , diameter D _n , area A _n , sum of distances between vertices S _n and width W _n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization methods. Numerous open problems are mentioned, as well as series of test problems for global optimization and non-linear programming codes.     

Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems

This paper is devoted to inequalities of Lieb-Thirring type. Let V be a nonnegative potential such that the corresponding Schrödinger operator has an unbounded sequence of eigenvalues (λi(V))_i∈N^∗. We prove that there exists a positive constant C(γ), such that, if γ>d/2, then

(∗)     

Frequency polygons for continuous random fields

We study the frequency polygon investigated by Scott (J Am Stat Assoc 80: 348–354, 1985) as a nonparametric density estimate for a continuous and stationary real random field \({\left( X_{\mathbf{t}},\mathbf{t}\in\mathbb{R}^{N}\right)}\). We establish the asymptotic expressions for the integrated pointwise squared bias and the integrated pointwise squared variance of the estimate when the field is observed over a rectangular domain of \({\mathbb{R}^{N}}\). Under mild mixing conditions, we show that the estimate achieves the same rate of convergence to zero of the integrated mean squared error as kernel estimators and it can also attain the optimal uniform strong rate of convergence \({\left(\widehat{\mathbf{T}}^{-1} \log \widehat{\mathbf{T}}\right)^{1/3}}\) for appropriate choices of the bin widths.     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

Diffuse reflection diameter and radius for convex-quadrilateralizable polygons

In this paper we study the diffuse reflection diameter and diffuse reflection radius problems for convex-quadrilateralizable polygons. In the usual model of diffuse reflection, a light ray incident at a point on the reflecting surface is reflected from that point in all possible inward directions. A ray reflected from a polygonal edge may graze that reflecting edge but an incident ray cannot graze the reflecting edge. The diffuse reflection diameter of a simple polygon $P$ is the minimum number of diffuse reflections that may be needed in the worst case to illuminate any target point $t$ from any point light source $s$ inside $P$. We show that the diameter is upper bounded by $\frac{3n?10}{4}$ in our usual model of diffuse reflection for convex-quadrilateralizable polygons. To the best of our knowledge, this is the first upper bound on diffuse reflection diameter within a fraction of $n$ for such a class of polygons. We also show that the diffuse reflection radius of a convex-quadrilateralizable simple polygon with $n$ vertices is at most $\frac{3n?10}{8}$ under the usual model of diffuse reflection. In other words, there exists a point inside such a polygon from which $\frac{3n?10}{8}$ usual diffuse reflections are always sufficient to illuminate the entire polygon. In order to establish these bounds for the usual model, we first show that the diameter and radius are $\frac{n?4}{2}$ and $?\frac{n?4}{4}?$ respectively, for the same class of polygons for a relaxed model of diffuse reflections; in the relaxed model an incident ray is permitted to graze a reflecting edge before turning and reflecting off the same edge at any interior point on that edge. We also show that the worst-case diameter and radius lower bounds of $\frac{n?4}{2}$ and $?\frac{n?4}{4}?$ respectively, are sometimes attained in the usual model, as well as in the relaxed model of diffuse reflection.     

Hypergraphs and abstract polygons

The notion of a graph has recently been generalized to include structures called hypergraphs which have two or more vertices per edge. A hypergraph is called 2-settled if each pair of distinct vertices is contained in at most one edge. A connected 2-settled hypergraph which has at least two edges through each vertex might be called an abstract polygon. Lemma: Every abstract polygon contains a cycle. Shephard and Coxeter have examined certain abstract polygons called regular complex polygons, each of which is denoted by a symbol p {q} r where there are p vertices on each edge and r edges through each vertex. Theorem: The girth of the non-starry regular complex polygon p {q} r is q. Thus, the number q is finally given a simple combinatoric interpretation.