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 -     

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.     

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.     

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.     

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.     

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

(∗)     

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.     

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.     

Untangling polygons and graphs

Untangling is a process in which some vertices of a plane graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph $C_n$ while keeping at least $\mathrm{\Omega }(n^{2/3})$ vertices fixed. For any graph G, we also present an upper bound for the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree and diameter of G. One of its consequences is the upper bound $O({(n\log n)}^{2/3})$ for all 3-vertex-connected planar graphs.     

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.     

Some extremal problems for circular polygons

The main result of this paper is the solution of the following problem posed by J. Hersch: find the maximal conformal radius on the family of all hyperbolic polygons with n sides (n ≥ 3). It is proved that the maximum is attained on a regular polygon. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993, pp. 127–136. Translated by A. Yu. Solynin.