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.     

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.     

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.     

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.     

Bonnesen-style inequalities and pseudo-perimeters for polygons

In this paper, we establish some Bonnesen-style isoperimetric inequalities for plane polygons via an analytic isoperimetric inequality and an isoperimetric inequality in pseudo-perimeters of polygons.1991 Mathematics Subject Classification 51M10, 51M25,52A40,26D10.     

Steiner minimal trees for regular polygons

Fifty years ago Jarnik and Kössler showed that a Steiner minimal tree for the vertices of a regularn-gon contains Steiner points for 3 n5 and contains no Steiner point forn=6 andn13. We complete the story by showing that the case for 7n12 is the same asn13. We also show that the set ofn equally spaced points yields the longest Steiner minimal tree among all sets ofn cocircular points on a given circle.     

Regular Steiner polygons

The symmetric difference area functional is minimized for a pair of planar convex polygons. Two solution procedures are outlined: a direct constructive methodology and a support function formulation. Examples illustrate the solution methodology.     

Recuttings of polygons

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 2, pp. 79–82, April–June, 1993.     

Algorithms for minimum length partitions of polygons

We show that it is possible to find a diagonal partition of anyn-vertex simple polygon into smaller polygons, each of at mostm edges, minimizing the total length of the partitioning diagonals, in timeO(n ³ m ²). We derive the same asymptotic upper time-bound for minimum length diagonal partitions of simple polygons into exactlym-gons provided that the input polygon can be partitioned intom-gons. Also, in the latter case, if the input polygon is convex, we can reduce the upper time-bound toO(n ³ logm).