Related papers: Multiple coverings with closed polygons

We demonstrate the existence of quasiconformal mappings on closed manifolds that cannot be decomposed as a composition of mappings with arbitrarily small conformal distortion.

We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results…

Polytope numbers for a polytope are a sequence of nonnegative integers that are defined by the facial information of a polytope. Every polygon is triangulable and a higher dimensional analogue of this fact states that every polytope is…

A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\Phi:X\to exp(X)$ such that $[x,y]\subset\Phi(x)\cup \Phi(y)$ for all $x,y\in X$. We prove that each convex subset of…

We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many…

Every polyhedron can be decomposed into a Minkowski sum (or vector sum) of a bounded polyhedron and a polyhedral cone. This paper establishes similar statements for some classes of discrete sets in discrete convex analysis, such as…

Suppose $\Delta$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $\Delta$ is at least $\binom{n}{c}-2c+1$,…

We present two algorithms for unfolding the surface of any polyhedron, all of whose faces are triangles, to a nonoverlapping, connected planar layout. The surface is cut only along polyhedron edges. The layout is connected, but it may have…

Let $K$ be a compact convex set and $m$ be a positive integer. The covering functional of $K$ with respect to $m$ is the smallest $\lambda\in[0,1]$ such that $K$ can be covered by $m$ translates of $\lambda K$. Estimations of the covering…

We address some questions concerning indecomposable polynomials and their spectrum. How does the spectrum behave via reduction or specialization, or via a more general ring morphism? Are the indecomposability properties equivalent over a…

A polynomial f (multivariate over a field) is decomposable if f = g(h) with g univariate of degree at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number…

In this work we study inside-out dissections of polygons and polyhedra. We first show that an arbitrary polygon can be inside-out dissected with $2n+1$ pieces, thereby improving the best previous upper bound of $4(n-2)$ pieces.…

Given a polygon $P$ in the plane, a {\em pop} operation is the reflection of a vertex with respect to the line through its adjacent vertices. We define a family of alternating polygons, and show that any polygon from this family cannot be…

Let $M$ be a convex cocompact acylindrical hyperbolic 3-manifold of infinite volume, and let $M^*$ denote the interior of the convex core of $M$. In this paper we show that any geodesic plane in $M^*$ is either closed or dense. We also show…

Both completely positive and completely copositive maps stay decomposable under tensor powers, i.e under tensoring the linear map with itself. But are there other examples of maps with this property? We show that this is not the case: Any…

Given a small polygon S, a big simple polygon B and a positive integer k, it is shown to be NP-hard to determine whether k copies of the small polygon (allowing translation and rotation) can be placed in the big polygon without overlap.…

Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…

One can define what it means for a compact manifold with corners to be a "contractible manifold with contractible faces." Two combinatorially equivalent, contractible manifolds with contractible faces are diffeomorphic if and only if their…

We prove that, for any two polyhedral manifolds $\mathcal P, \mathcal Q$, there is a polyhedral manifold $\mathcal I$ such that $\mathcal P, \mathcal I$ share a common unfolding and $\mathcal I,\mathcal Q$ share a common unfolding. In other…

We prove that any convex body in the plane can be partitioned into $m$ convex parts of equal areas and perimeters for any integer $m\ge 2$; this result was previously known for prime powers $m=p^k$. We also discuss possible…