Affinely Regular Polygons as Extremals of Area Functionals

For any convex n-gon P we consider the polygons obtained by dropping a vertex or an edge of P. The area distance of P to such (n−1)-gons, divided by the area of P, is an affinely invariant functional on n-gons whose maximizers coincide with the affinely regular polygons. We provide a complete proof of this result. We extend these area functionals to planar convex bodies and we present connections with the affine isoperimetric inequality and parallel X-ray tomography.     

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).     

A Discrete Isoperimetric Problem

We prove that the perimeter of any convex n-gons of diameter 1 is at most n2nsin (/2n). Equality is attained here if and only if n has an odd factor. In the latter case, there are (up to congruence) only finitely many extremal n-gons. In fact, the convex n-gons of diameter 1 and perimeter n2n sin (/2n) are in bijective correspondence with the solutions of a diophantine problem.     

Una proprietà di massimo dei poligoni affinemente regolari

We consider an affinely invariant functional on the classP _m of the plane convex polygons, withm sides,m>4. We give, for allm, a geometrical property of the maximizing polygons. Form=5, 6 the maximum is attained if and only if the polygons are affinely regular. The maximum inP _2n is a bound of another functional which arises in reconstructing plane convex bodies from the length of its chords alongn directions.     

The robust minimum spanning tree problem: Compact and convex uncertainty

We consider the robust minimum spanning tree problem where edges costs are on a compact and convex subset of Rⁿ. We give the location of the robust deviation scenarios for a tree and characterizations of strictly strong edges and non-weak edges leading to recognition algorithms.     

Flipping Edges in Triangulations

In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least edges that can be flipped. We also prove that O(n + k ² ) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T' such that to transform T into T' requires Ω(n ² ) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips. Received May 13, 1997, and in revised form July 21, 1998, and February 1, 1999.     

Convex bodies with extremal volumes having prescribed brightness in finitely many directions

We consider the class of convex bodies in ⁿ with prescribed projection (n – 1)-volumes along finitely many fixed directions. We prove that in such a class there exists a unique body (up to translation) with maximumn-volume. The maximizer is a centrally symmetric polytope and the normal vectors to its facets depend only on the assigned directions.Conditions for the existence of bodies with minimumn-volume in the class defined above are given. Each minimizer is a polytope, and an upper bound for the number of its facets is established.Work partially supported by Istituto di Analisi Globale e Applicazioni, CNR, Firenze.     

THE MINIMUM AREA OF A SIMPLE POLYGON WITH GIVEN SIDE LENGTHS

Answering a question of H. Harborth, for any given a ₁,...,a _n > 0, satisfying we determine the infimum of the areas of the simple n-gons in the Euclidean plane, having sides of length a ₁,...,a _n (in some order). The infimum is attained (in limit) if the polygon degenerates into a certain kind of triangle, plus some parts of zero area. We show the same result for simple polygons on the sphere (of not too great length), and for simple polygons in the hyperbolic plane. Replacing simple n-gons by convex ones, we answer the analogous questions. The infimum is attained also here for degeneration into a certain kind of triangle. This revised version was published online in August 2006 with corrections to the Cover Date.     

Poisson Convergence in the n-Cube

We consider two types of random subgraphs of the n-cube Q_n obtained by independent deletion the vertices (together with all edges incident with them) or the edges of Qⁿ, respectively, with a prescribed probability q = 1 — p. For these two probabilistic models we determine some values of the probability p for which the number of (isolated) k-dimensional subcubes or the number of vertices of a given degree k, respectively, has asymptotically a Poisson or a Normal distribution. The technique which will be used is that of Poisson convergence introduced by BARBOUR [1] (see also [4]).     

Which n-Venn Diagrams Can Be Drawn with Convex k-Gons?

We establish a new lower bound for the number of sides required for the component curves of simple Venn diagrams made from polygons. Specifically, for any n-Venn diagram of convex k-gons, we prove that k ≥ (2ⁿ - 2 - n) / (n (n - 2)). In the process we prove that Venn diagrams of seven curves, simple or not, cannot be formed from triangles. We then give an example achieving the new lower bound of a (simple, symmetric) Venn diagram of seven convex quadrilaterals. Previously Grunbaum had constructed a symmetric 7-Venn diagram of non-convex 5-gons ["Venn Diagrams II", Geombinatorics 2:25-31, 1992].     

Steiner polygons in the Steiner problem

A polygon, whose vertices are points in a given setA ofn points, is defined to be a Steiner polygon ofA if all Steiner minimal trees forA lie in it. Cockayne first found that a Steiner polygon can be obtained by repeatedly deleting triangles from the boundary of the convex hull ofA. We generalize this concept and give a method to construct Steiner polygons by repeatedly deletingk-gons,k n. We also prove the uniqueness of Steiner polygons obtained by our method.     

The Minimum Area of Convex Lattice <Emphasis Type="Italic">n</Emphasis>-Gons

Let A(n) be the minimum area of convex lattice n-gons. We prove that lim A(n)/n³ exists. Our computations suggest that the value of the limit is very close to 0.0185067....* Partially supported by Hungarian National Science Foundation Grants T 032452 and T 029255.     

Ramsey-Type Results for Geometric Graphs, II

We show that for any two-coloring of the segments determined by n points in the plane, one of the color classes contains noncrossing cycles of lengths . This result is tight up to a multiplicative constant. Under the same assumptions, we also prove that there is a noncrossing path of length Ω(n ^2/3 ) , all of whose edges are of the same color. In the special case when the n points are in convex position, we find longer monochromatic noncrossing paths, of length . This bound cannot be improved. We also discuss some related problems and generalizations. In particular, we give sharp estimates for the largest number of disjoint monochromatic triangles that can always be selected from our segments. Received March 25, 1997, and in revised form March 5, 1998.     

Bicolored Matchings in Some Classes of Graphs

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R|=n+1 such that perfect matchings with k red edges exist for all k,0≤k≤n. Given two integers p<q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p≤4 there is a set R with |R|=p for which perfect matchings M_k exist with |M_k∩R|≤k for all k≤p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k.     

Filling of a given boundary by p-gons and related problems

We consider here (p,s)-polycycles (3ps) i.e. plane graphs, such that all interior faces are p-gons, all interior vertices are s-valent and any vertex of the boundary (i.e. the exterior face) has valency within [2,s]. The boundary sequence of a (p,s)-polycycle P is the sequence b(P) enumerating, up to a cyclic shift or reversal, the consecutive valencies of vertices of the boundary. We show that the values p=3,4 are the only ones, such that the boundary sequence defines its (p,3)-filling (i.e. a (p,3)-polycycle with given boundary) uniquely.Also we give new results in the enumeration of maps M_n(p,q) (i.e. plane 3-valent maps with only p- and q-gonal faces, such that the q-gons are organized in an n-ring) and two of their generalizations.Both problems are similar (3-valent filling by p-gons of a boundary or of a ring of q-gons) and the same programs were used for both computations.     

Rotatingn-gon/kn-gon vortex configurations

Summary We demonstrate the existence of stationary point-vortex configurations consisting ofk vortexn-gons and a vortexkn-gon. These configurations exist only for specific values of the vortex strengths; the relative vortex strengths of such a consiguration can be uniquely expressed as functions of the radii of the polygons. Thekn-gon must be oriented so as to be fixed by any reflection fixing one of then-gons; for sufficiently smallk, we show that then-gons must be oriented in such a way that the entire configuration shares the symmetries of any of then-gons. Necessary conditions for the formal stability of general stationary point-vortex configurations set conditions on the vortex strengths. We apply these conditions to then-gon/kn-gon configurations and carry out a complete linear and formal stability analysis in the casek=n=2, showing that linearly and nonlinearly orbitally stable configurations exist.     

Cyclic polygons with rational sides and area

We generalise the notion of Heron triangles to rational-sided, cyclic n-gons with rational area using Brahmagupta's formula for the area of a cyclic quadrilateral and Robbins' formulæ for the area of cyclic pentagons and hexagons. We use approximate techniques to explore rational area n-gons for n greater than six. Finally, we produce a method of generating non-Eulerian rational area cyclic n-gons for even n and conjecturally classify all rational area cyclic n-gons.     

Nonnegative inverse eigenvalue problems with partial eigendata

In this paper we consider the inverse problem of constructing an n × n real nonnegative matrix A from the prescribed partial eigendata. We first give the solvability conditions for the inverse problem without the nonnegative constraint and then discuss the associated best approximation problem. To find a nonnegative solution, we reformulate the inverse problem as a monotone complementarity problem and propose a nonsmooth Newton-type method for solving its equivalent nonsmooth equation. Under some mild assumptions, the global and quadratic convergence of our method is established. We also apply our method to the symmetric nonnegative inverse problem and to the cases of prescribed lower bounds and of prescribed entries. Numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings.     

Approximation for Minimum Triangulations of Simplicial Convex 3-Polytopes

A minimum triangulation of a convex 3-polytope is a triangulation that contains the minimum number of tetrahedra over all its possible triangulations. Since finding minimum triangulations of convex 3-polytopes was recently shown to be NP-hard, it becomes significant to find algorithms that give good approximation. In this paper we give a new triangulation algorithm with an improved approximation ratio 2 - Ω(1/\sqrt n ) , where n is the number of vertices of the polytope. We further show that this is the best possible for algorithms that only consider the combinatorial structure of the polytopes. Received August 5, 2000, and in revised form March 29, 2001, and May 3, 2001. Online publication October 12, 2001.