If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex polytope of dimension $n-3$, each facet of which consists of those $n$-gons in which the length of a particular edge has fallen to zero. Bavard and Ghys require unit area instead, which gives them a hyperbolic polytope. Those two polytopes are combinatorially equivalent, so either is fine for our purposes.
Such a fixed-angles polytope is combinatorially richer when the angles are well balanced. We say that fixed external angles are "majority dominant" when every consecutive string of more than half of them sums to more than $π$. When $n$ is odd, we show that the fixed-angles polytope for any majority-dominant angles is dual to the cyclic polytope $C_{n-3}(n)$. To extend that result to even $n$, we require that the angles also have "dipole tie-breaking": None of the $n$ strings of length $n/2$ sums to precisely $π$, and the $n/2$ that sum to more than $π$ overlap as much as possible, all containing a particular angle.
Fixing the vertex angles is uncommon, however; people more often fix the edge lengths. That is harder, in part because fixed-lengths $n$-gons may not be convex, but mostly because fixing the lengths constrains the angles nonlinearly -- so the resulting moduli spaces, called "polygon spaces", are curved. Using Schwarz-Christoffel maps, Kapovich and Millson show that the subset of that polygon space in which the $n$-gons are convex and traversed counterclockwise is homeomorphic to the fixed-angles polytope above, for those same fixed values. Each such subset is thus a topological polytope; and it is dual cyclic whenever the fixed lengths are majority dominant and, for even $n$, have dipole tie-breaking.
We show that every heptagon is a section of a $3$-polytope with $6$ vertices. This implies that every $n$-gon with $n\geq 7$ can be obtained as a section of a $(2+\lfloor\frac{n}{7}\rfloor)$-dimensional polytope with at most $\lceil\frac{6n}{7}\rceil$ vertices; and provides a geometric proof of the fact that every nonnegative $n\times m$ matrix of rank $3$ has nonnegative rank not larger than $\lceil\frac{6\min(n,m)}{7}\rceil$. This result has been independently proved, algebraically, by Shitov (J. Combin. Theory Ser. A 122, 2014).
Let $S$ be a set of $2n$ points on a circle such that for each point $p \in S$ also its antipodal (mirrored with respect to the circle center) point $p'$ belongs to $S$. A polygon $P$ of size $n$ is called \emph{antipodal} if it consists of precisely one point of each antipodal pair $(p,p')$ of $S$. We provide a complete characterization of antipodal polygons which maximize (minimize, respectively) the area among all antipodal polygons of $S$. Based on this characterization, a simple linear time algorithm is presented for computing extremal antipodal polygons. Moreover, for the generalization …
A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such $n$-gons are parametrized by the subspaces of $(0,\infty)^n$ that contain their side length collections, and area and circumcircle or "collar" radius determine symmetric, smooth functions on these spaces. We give formulas for and bounds on the derivatives of these functions, and make some observations on their behavior. Notably, the monotonicity properties of area and circumcircle …
In his works [R1,R2] David Robbins proposed several interrelated conjectures on the area of the polygons inscribed in a circle as an algebraic function of its sides. Most recently, these conjectures have been established in the course of several independent investigations. In this note we give an informal outline of these developments.
We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 0< j <k we prove that the maximum possible number of j-dimensional faces of a centrally symmetric d-dimensional polytope with n vertices is at least (c_j(d)+o(1)) {n \choose j+1} for some c_j(d)>0 and at most (1-2^{-d}+o(1)){n \choose j+1}…
It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper, we announce an explicit formula of the Morse index for the signed area of a cyclic configuration. It depends not only on the combinatorics of a cyclic configuration, but also includes some metric characterization.
Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.
In this paper, we study dilation of cyclic polytopes with the vertices defined by a generator of the simplest cubic fields. In particular, for a specific range of values, we give a precise number of the contained lattice points.
In this paper we study the operation of cutting off edges of a simple $3$-polytope $P$ along the graph $Γ$. We give the criterion when the resulting polytope is simple and when it is flag. As a corollary we prove the analog of Eberhard's theorem about the realization of polygon vectors of simple $3$-polytopes for flag polytopes.
This paper presents a general method for obtaining radius of the corresponding circumference to a cyclical polygon $n$ sides given the lengths of said sides, using the notion of complex number. As of radius $r$, obtained, can then be calculated polygon area in question applying known expressions which requires, after $n=4$, a powerful calculation tool. Are also given like elements regarding non convex polygons cyclic, although in this respect it deepens less and is not considered, course, the calculation of areas.
We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide exact and asymptotic formulas describing a random realizable configuration, obtained either by sampling the points uniformly at random on the circle or by sampling a realizable configuration uniformly at random.
The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of flip graphs on pseudoline arrangements with contacts supported by a given sorting network. In this paper, we construct the brick polytope of a sorting network, obtained as the convex hull of the brick vectors associated to each pseudoline arrangement supported by the network. We combinatorially characterize the vertices of this polyt…
It is well known that Heron's theorem provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its sides. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges, which, surprisingly, has revealed to be far from simple. In this paper we investigate such a problem by following a new and elementary approach. We start from the simple observation that the incircle of a right triangle touches its hypothen…
We provide a new proof of the elementary geometric theorem on the existence and uniqueness of cyclic polygons with prescribed side lengths. The proof is based on a variational principle involving the central angles of the polygon as variables. The uniqueness follows from the concavity of the target function. The existence proof relies on a fundamental inequality of information theory. We also provide proofs for the corresponding theorems of spherical and hyperbolic geometry (and, as a byproduct, in $1+1$ spacetime). The spherical theorem is reduced to the euclidean one. The proof of the hyperb…
An equidistant polytope is a special equidistant set in the space $\mathbb{R}^n$ all of whose boundary points have equal distances from two finite systems of points. Since one of the finite systems of the given points is required to be in the interior of the convex hull of the other one we can speak about inner and outer focal points of the equidistant polytope. It is of type $(q, p)$, where $q$ is the number of the outer focal points and $p$ is the number of the inner focal points. The equidistancy is the generalization of convexity because a convex polytope can be given as an equidistant pol…
The Erdos-Szekeres theorem states that for any natural k there is a natural number g(k) such that any set of at least g(k) points on a plane in general position contains a set of k points that are the extreme points of a convex polytope. We generalize and refine this theorem, having the general-position condition removed and a convex polygon defined as an ordered sequence of points such that the union of the edges of the polygon coincides with the boundary of its convex hull.
For a convex body $K \subset {\mathbb R}^n$, let $K^z = \{y\in{\mathbb R}^n : \langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$ be the polar body of $K$ with respect to the center of polarity $z \in {\mathbb R}^n$. The goal of this paper is to study the maximum of the volume product $\mathcal{P}(K)=\min_{z\in {\rm int}(K)}|K||K^z|$, among convex polytopes $K\subset {\mathbb R}^n$ with a number of vertices bounded by some fixed integer $m \ge n+1$. In particular, we prove that the supremum is reached at a simplicial polytope with exactly $m$ vertices and we provide a new proof of a res…
We investigate the ``generalized Heron polynomial'' that relates the squared area of an n-gon inscribed in a circle to the squares of its side lengths. For a (2m+1)-gon or (2m+2)-gon, we express it as the defining polynomial of a certain variety derived from the variety of binary (2m-1)-forms having m-1 double roots. Thus we obtain explicit formulas for the areas of cyclic heptagons and octagons, and illuminate some mysterious features of Robbins' formulas for the areas of cyclic pentagons and hexagons. We also introduce a companion family of polynomials that relate the squared area of an n-go…
We define and study a variant of the center of mass of a polygon and, more generally, of a simplicial polytope which we call the Circumcenter of Mass (CCM). The Circumcenter of Mass is an affine combination of the circumcenters of the simplices in a triangulation of a polytope, weighted by their volumes. For an inscribed polytope, CCM coincides with the circumcenter. Our motivation comes from the study of completely integrable discrete dynamical systems, where the CCM is an invariant of the discrete bicycle (Darboux) transformation and of recuttings of polygons. We show that the CCM satisfies …
In 1980, V. I. Arnold studied the classification problem for convex lattice polygons of given area. Since then, this problem and its analogues have been studied by many authors, including $\mathrm{B\acute{a}r\acute{a}ny}$, Lagarias, Pach, Santos, Ziegler and Zong. Recently, Zong proposed two computer programs to prove Hadwiger's covering conjecture and Borsuk's partition problem, respectively, based on enumeration of the convex lattice polytopes contained in certain balls. For this purpose, similar to $\mathrm{B\acute{a}r\acute{a}ny}$ and Pach's work on volume and Liu and Zong's work on cardin…
These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f-vectors, and extremal constructions. The first lecture treats 3-dimensional polytopes; it includes a complete proof of the Koebe--Andreev--Thurston theorem, using the variational principle by Bobenko & Springborn (2004). In Lecture 2 we look at f-vector shapes of very high-dimensional polytopes. The third lecture explains a surprisingly simple construction for 2-simple 2-simplicial 4-polytopes, which have symmetric f-vectors. Lecture 4 sketches the geometry of …
In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, the only known polytopes (besides polygons) that are edge-transitive without being vertex-transitive. We show that these polyhedra do not have higher-dimensional analogues, that is, that in dimension $d\ge 4$, edge-transitivity of convex polytopes implies vertex-transitivity. More generally, we give a classification of all convex polytopes which at the same time have all edges of the same length, an edge in-sphere and a bipartite edge-graph. We show that any such po…
The solid-angle sum $A_{\mathcal{P}} (t)$ of a rational polytope ${\mathcal{P}} \subset \mathbb{R}^d$, with $t \in \mathbb{Z}$ was first investigated by I.G. Macdonald. Using our Fourier-analytic methods, we are able to establish an explicit formula for $A_{\mathcal{P}} (t)$, for any real dilation $t$ and any rational polygon ${\mathcal{P}} \subset \mathbb{R}^2$. Our formulation sheds additional light on previous results, for lattice-point enumerating functions of triangles, which are usually confined to the case of integer dilations. Our approach differs from that of Hardy and Littlewood in 1…
In 1980, V.I. Arnold studied the classification problem for convex lattice polygons of given area. Since then this problem and its analogues have been studied by B'ar'any, Pach, Vershik, Liu, Zong and others. Upper bounds for the numbers of non-equivalent ddimensional convex lattice polytopes of given volume or cardinality have been achieved. In this paper, by introducing and studying the unimodular groups acting on convex lattice polytopes, we obtain lower bounds for the number of non-equivalent d-dimensional convex lattice polytopes of bounded volume or given cardinality, which are essential…
We construct, for any positive integer n, a family of n congruent convex polyhedra in R^3, such that every pair intersects in a common facet. Previously, the largest such family contained only eight polytopes. Our polyhedra are Voronoi regions of evenly distributed points on the helix (t, cos t, sin t). With a simple modification, we can ensure that each polyhedron in the family has a point, a line, and a plane of symmetry. We also generalize our construction to higher dimensions and introduce a new family of cyclic polytopes.
Although previous research has found several facts concerning chord lengths of regular polytopes, none of these investigations has considered whether any of these facts define relationships that might generalize to the chord lengths of all regular polytopes. Consequently, this paper explores whether four findings of previous studies-- viz., the four facts relating to the sums and products of squared chord lengths of regular polygons inscribed in unit circles-- can be generalized to all regular ($n$-dimensional) polytopes (inscribed in unit $n$-spheres). We show that (a) one of these four facts…
Given the n vertices of a convex polygon in cyclic order, can the triangle of maximum area inscribed in P be determined by an algorithm with O(n) time complexity? A purported linear-time algorithm by Dobkin and Snyder from 1979 has recently been shown to be incorrect by Keikha, Löffler, Urhausen, and van der Hoog. These authors give an alternative algorithm with O(n log n) time complexity. Here we give an algorithm with linear time complexity.
The circumcircle of a planar convex polygon P is a circle C that passes through all vertices of P. If such a C exists, then P is said to be cyclic. Fix C to have unit radius. While any two angles of a uniform cyclic triangle are negatively correlated, any two sides are independent. In contrast, for a uniform cyclic quadrilateral, any two sides are negatively correlated, whereas any two adjacent angles are uncorrelated yet dependent.
A set $Ω\subset \mathbb{R}^d$ is said to be spectral if the space $L^2(Ω)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by translations. This suggests a conjecture that a product set $Ω= A \times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result i…
