Related papers: Polygon Convexity: Another O(n) Test

For every $n \geq 5$, we show that the Kneser graph of triangulations of a convex $n$-gon contains a Hamiltonian cycle.

A convex polygon $Q$ is inscribed in a convex polygon $P$ if every side of $P$ contains at least one vertex of $Q$. We present algorithms for finding a minimum area and a minimum perimeter convex polygon inscribed in any given convex…

The present article includes the enumeration of $n$-polygons with a certain symmetry property: For an even number $n$ of vertices, we count the $n$-polygons with $\frac{n}{2}$ symmetry axes. In addition, if $n$ is a power of 2, we show the…

An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron…

When the number of non-triangular faces adjacent to a vertex $v$ is less than or equal to three, the vertex $v$ will be called (\emph{combinatorially}) \emph{rigid}. We study the number of rigid vertices and suggest a conjecture on a…

We revisit a standard polygon containment problem: given a convex $k$-gon $P$ and a convex $n$-gon $Q$ in the plane, find a placement of $P$ inside $Q$ under translation and rotation (if it exists), or more generally, find the largest copy…

In a convex n-gon, let d[1] > d[2] > ... denote the set of all distances between pairs of vertices, and let m[i] be the number of pairs of vertices at distance d[i] from one another. Erdos, Lovasz, and Vesztergombi conjectured that m[1] +…

Our main theoretical result is that, if a simple polytope has a pair of complementary vertices (i.e., two vertices with no facets in common), then it has at least two such pairs, which can be chosen to be disjoint. Using this result, we…

Fix an integer n>=1. Suppose that a simple polygon is the union of n triangles whose vertices along the common boundary are arranged cyclically. How many sides can such a union -- to be called regular -- have at most? This gives OEIS…

A simple $n$-gon is a polygon with $n$ edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass asked the following question: For $n \geq 5$ odd, what is the maximum perimeter of a…

This paper proposed a method to judge whether the point is inside or outside of the simple convex polygon by the intersection of the vertical line. It determined the point to an area enclosed by two straight lines, then convert the problem…

For any finite set $\A$ of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set $\A$, where a marked graph is defined as a…

Let $S\subset \mathbb{R}^d$ $(d\geq 2)$. A set $S$ is said to be $m$-point convex, if for every $m$ distinct points in $S$, at least one of the line-segments determined by them lies in $S$. We also say that $S$ has property $P_m$. Let…

Let $P$ be a convex polyhedron in $\mathbb{R}^3$. The skeleton of $P$ is the graph whose vertices and edges are the vertices and edges of $P$, respectively. We prove that, if these vertices are on the unit-sphere, the skeleton is a $(0.999…

Let $D$ be a connected oriented graph. A set $S \subseteq V(D)$ is convex in $D$ if, for every pair of vertices $x, y \in S$, the vertex set of every $xy$-geodesic, ($xy$ shortest directed path) and every $yx$-geodesic in $D$ is contained…

According to Suk's breakthrough result on the Erdos-Szekeres problem, any point set in general position in the plane, which has no $n$ elements that form the vertex set of a convex $n$-gon, has at most $2^{n+O\left({n^{2/3}\log n}\right)}$…

The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve $\pi$ is $c$-packed if the length of the curve lying inside any ball is at most $c$ times the radius of the ball, and its congestion is…

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 >= (2^n - 2 - n) / (n (n-2)). In the…

Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$,…

We study convex cyclic polygons, that is, inscribed $n$-gons. Starting from P. Schreiber's idea, published in 1993, we prove that these polygons are not constructible from their side lengths with straightedge and compass, provided $n$ is at…