Related papers: On the Poncelet triangle condition over finite fie…
Solutions of a variational inequality are found by giving conditions for the monotone convergence with respect to a cone of the Picard iteration corresponding to its natural map. One of these conditions is the isotonicity of the projection…
We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted…
In this paper, we deal with the question; under what conditions the points $P_i(xi,yi)$ $(i = 1,\cdots, n)$ form a convex polygon provided $x_1 < \cdots < x_n$ holds. One of the main findings of the paper can be stated as follows: "Let…
Nandakumar asked whether there is a tiling of the plane by pairwise non-congruent triangles of equal area and equal perimeter. Here a weaker result is obtained: there is a tiling of the plane by pairwise non-congruent triangles of equal…
We review a certain problem on covering triangles in the plane. Equivalently, it can be viewed as a family of 'isobilliard' inequalities in convex shapes, and as a special case of Viterbo's conjecture in symplectic geometry. We give an…
We consider an even probability distribution on the $d$-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given $N$ independent random vectors with this distribution, under the…
Working over a field of characteristic other than $2$, we examine a relationship between quadrilaterals and the pencil of conics passing through their vertices. Asymptotically, such a pencil of conics is what we call a bisector field, a set…
We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that…
Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we study a two-parameter family of roughly hexagonal regions in the hexagonal grid and…
Let $P$ be a finite point set in the plane. A \emph{$c$-ordinary triangle} in $P$ is a subset of $P$ consisting of three non-collinear points such that each of the three lines determined by the three points contains at most $c$ points of…
We consider the wave equation $(\partial_t^2-\Delta)u=0$ on a planar triangular domain $\Omega\subset\mathbb{R}^2$ with Dirichlet boundary conditions. We use a commutator and integration by parts argument similar to that in…
In this paper, we consider the problem of determining in polynomial time whether a given planar point set $P$ of $n$ points admits 4-connected triangulation. We propose a necessary and sufficient condition for recognizing $P$, and present…
Let the Euclidean plane be simultaneously and independently endowed with a Poisson point process and a Poisson line process, each of unit intensity. Consider a triangle T whose vertices all belong to the point process. The triangle is…
Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In…
Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability that a random triangle on a flat torus is homotopically trivial. Our main tool for this…
There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is…
In a previous paper, we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the…
We study the asymptotic proportion of smooth plane curves over a finite field $\mathbb{F}_q$ which are tangent to every line defined over $\mathbb{F}_q$. This partially answers a question raised by Charles Favre. Our techniques include…
Let $S$ be a set of $n$ points in general position in the plane. The Second Selection Lemma states that for any family of $\Theta(n^3)$ triangles spanned by $S$, there exists a point of the plane that lies in a constant fraction of them.…
We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic…