Related papers: Skeletal Cut Loci on Convex Polyhedra
Base polytopes of polymatroids, also known as generalized permutohedra, are polytopes whose edges are parallel to a vector of the form $\mathbf{e}_i - \mathbf{e}_j$. We consider the following computational problem: Given two vertices of a…
The aim of this paper is to determine the structure of the cut locus for a class of surfaces of revolution homeomorphic to a cylinder. Let $M$ denote a cylinder of revolution which admits a reflective symmetry fixing a parallel called the…
We showed in another paper [arXiv:1103.1759] that every connected graph can be realized as the cut locus of some point on some riemannian surface $S$. Here, criteria for the orientability of $S$ are given, and are applied to classify the…
An unzipping of a polyhedron P is a cut-path through its vertices that unfolds P to a non-overlapping shape in the plane. It is an open problem to decide if every convex P has an unzipping. Here we show that there are nearly flat convex…
We extend the notion of a star unfolding to be based on a simple quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple, planar polygon: shortest paths from…
This paper presents a selected tour through the theory and applications of lifts of convex sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original set. Many convex sets have lifts that are…
We consider a natural variation of the concept of stabbing a segment by a simple polygon: a segment is stabbed by a simple polygon $\mathcal{P}$ if at least one of its two endpoints is contained in $\mathcal{P}$. A segment set $S$ is…
We study the problem of partitioning a polygon into the minimum number of subpolygons using cuts in predetermined directions such that each resulting subpolygon satisfies a given width constraint. A polygon satisfies the unit-width…
Globally non-positively curved, or CAT(0), polyhedral complexes arise in a number of applications, including evolutionary biology and robotics. These spaces have unique shortest paths and are composed of Euclidean polyhedra, yet many…
We show that the minimum distance projection in the L1-norm from an interior point onto the boundary of a convex set is achieved by a single, unidimensional projection. Application of this characterization when the convex set is a…
The convex hull peeling of a point set consists in taking the convex hull, then removing the extreme points and iterating that procedure until no point remains. The boundary of each hull is called a layer. Following on from [15], we study…
Let $\Pi$ be a convex decomposition of a set $P$ of $n\geq 3$ points in general position in the plane. If $\Pi$ consists of more than one polygon, then either $\Pi$ contains a deletable edge or $\Pi$ contains a contractible edge.
Skeletal polyhedra and polygonal complexes are finite or infinite periodic structures in 3-space with interesting geometric, combinatorial, and algebraic properties. These structures can be viewed as finite or infinite periodic graphs…
In this article, we investigate the cut locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. We explore the deformation and characterization of the cut locus, extending the results of Basu and the…
M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for…
We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices $xx^{\mathrm{H}}$, where the elements of $x \in \mathbb{C}^n$ are $m$th unit roots. These polytopes have applications in ${\text{MAX-3-CUT}}$, digital…
A closed quasigeodesic is a closed curve on the surface of a polyhedron with at most $180^\circ$ of surface on both sides at all points; such curves can be locally unfolded straight. In 1949, Pogorelov proved that every convex polyhedron…
The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in R^3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron…
Let $\Omega$ be an open set in $\mathbb{R}^n$ with $C^1$-boundary and $\Sigma$ be the skeleton of $\Omega$, which consists of points where the distance function to $\partial\Omega$ is not differentiable. This paper characterizes the cut…
We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other…