Related papers: Two Regular Polygons with a Shared Vertex

For two points in the closure of a simple polygon $P$, we say that they see each other in $P$ if the line segment uniting them does not intersect the exterior of $P$. The visibility graph of $P$ is the graph whose vertex set is the vertex…

We investigate some combinatorial properties of convex polytopes simple in edges. For polytopes whose nonsimple vertices are located sufficiently far one from another, we prove an analog of the Hard Lefschetz theorem. It implies Stanley's…

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…

A seminal theorem of Tverberg states that any set of $T(r,d)=(r-1)(d+1)+1$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Almost any collection of fewer points in…

We study the space of orthogonally additive $n$-homogeneous polynomials on $C(K)$. There are two natural norms on this space. First, there is the usual supremum norm of uniform convergence on the closed unit ball. As every orthogonally…

By a polygonization of a finite point set $S$ in the plane we understand a simple polygon having $S$ as the set of its vertices. Let $B$ and $R$ be sets of blue and red points, respectively, in the plane such that $B\cup R$ is in general…

Angular equivalence is introduced and shown to be an equivalence relation among the norms on a fixed real vector space. It is a finer notion than the usual (topological) notion of norm equivalence. Angularly equivalent norms share certain…

The article studies a generalization of the classical Fermat-Torricelli problem to normed spaces of arbitrary finite dimension. Necessary and sufficient conditions for the uniqueness of the solution of the Fermat-Torricelli problem for any…

Wythoff's construction associates a uniform polytope to a Coxeter diagram whose vertices are decorated with crosses, which indicate the subgroup stabilizing a generic point. Champagne, Kjiri, Patera, and Sharp remarked that by associating…

We present an algorithm for constructing the fixed point of a general non-isometric similarity of the plane.

We introduce a new type of mappings in metric spaces which are three-point analogue of the well-known Kannan type mappings and call them generalized Kannan type mappings. It is shown that in general case such mappings are discontinuous but…

Let $S$ be a set of $n$ points in general position in the plane, and let $X_{k,\ell}(S)$ be the number of convex $k$-gons with vertices in $S$ that have exactly $\ell$ points of $S$ in their interior. We prove several equalities for the…

It is conjectured since long that for any convex body $P\subset \mathbb{R}^n$ there exists a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $P$. The conjecture is known to be true for…

In this note, we completely describe the shape of the bisector of two given points in a two-dimensional normed vector space. More precisely, we show that, depending on the position of two given points with respect to the shape of the unit…

Motivated by a question of R.\ Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex quadrangles of the same area and the same perimeter. As a byproduct we obtain vertex-to-vertex dissections of the…

In this paper, we show that if we decompose a polygon into two smaller polygons, then by comparing the number of extremal vertices in the original polygon versus the sum of the two smaller polygons, we can gain at most two globally extremal…

In this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime. The proof is based on both variational and geometric arguments involving the causal structure of…

We introduce a new type of mappings in metric space which are three-point analogue of the well-known Chatterjea type mappings, and call them generalized Chatterjea type mappings. It is shown that such mappings can be discontinuous as is the…

Consider a bicolored point set $P$ in general position in the plane consisting of $n$ blue and $n$ red points. We show that if a subset of the red points forms the vertices of a convex polygon separating the blue points, lying inside the…

Two planar graphs G1 and G2 sharing some vertices and edges are `simultaneously planar' if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether…