Related papers: The Number of Triangles Needed to Span a Polygon E…

We study $d$-dimensional simplicial complexes that are PL embeddable in $\mathbb{R}^{d+1}$. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic…

Let $n$ be a positive integer, not a power of two. A \textit{Reinhardt polygon} is a convex $n$-gon that is optimal in three different geometric optimization problems: it has maximal perimeter relative to its diameter, maximal width…

Let $S$ be a set of $n$ points in the plane, $\wp(S)$ be the set of all simple polygons crossing $S$, $\gamma_P$ be the maximum angle of polygon $P \in \wp(S)$ and $\theta =min_{P\in\wp(S)} \gamma_P$. In this paper, we prove that…

Let $\Poly$ be a simple polygon with $n$ vertices. The \emph{dual graph} $\triang^*$ of a triangulation~$\triang$ of~$\Poly$ is the graph whose vertices correspond to the bounded faces of $\triang$ and whose edges connect those faces…

Any two triangulations of a closed surface with the same number of vertices can be transformed into each other by a sequence of regular flips, provided the number of vertices exceeds a number N depending on the surface. Examples show that…

Motivated by classical work of Alon and F\"uredi, we introduce and address the following problem: determine the minimum number of affine hyperplanes in $\mathbb{R}^d$ needed to cover every point of the triangular grid $T_d(n) :=…

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible…

We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/sqrt(d) pivot steps on any simple d-polytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a…

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…

Every surface that is intrinsically polyhedral can be represented by a portalgon: a collection of polygons in the Euclidean plane with some pairs of equally long edges abstractly identified. While this representation is arguably simpler…

An arrangement of circles in which circles intersect only in angles of $\pi/2$ is called an \emph{arrangement of orthogonal circles}. We show that in the case that no two circles are nested, the intersection graph of such an arrangement is…

More than 25 years ago Chazelle~\emph{et al.} (FOCS 1991) studied the following question: Is it possible to cut any set of $n$ lines in ${\Bbb R}^3$ into a subquadratic number of fragments such that the resulting fragments admit a depth…

We prove that for every surface $\Sigma$ of Euler genus $g$, every edge-maximal embedding of a graph in $\Sigma$ is at most $O(g)$ edges short of a triangulation of $\Sigma$. This provides the first answer to an open problem of Kainen…

A result due to Burago and Zalgaller (1960, 1995) states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space $\mathbb{E}^3$. A…

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…

We give some new advances in the research of the maximum number of triangles that we may obtain in a simple arrangements of n lines or pseudo-lines.

Assume we are given a set of parallel line segments in the plane, and we wish to place a point on each line segment such that the resulting point set maximizes or minimizes the area of the largest or smallest triangle in the set. We analyze…

We consider triangle faced convex polyhedra inscribed in the unit sphere $S^2$ in ${\Bbb{R}}^3$. One way of measuring their deviation from regular polyhedra with triangular faces is to consider the quotient of the lengths of the longest and…

We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genus-surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most…

We consider the problem of computing the second elementary symmetric polynomial S^2_n(X) using depth-three arithmetic circuits of the form "sum of products of linear forms". We consider this problem over several fields and determine EXACTLY…