Related papers: On Flat Polyhedra deriving from Alexandrov's Theor…
Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a…
In this paper we discuss the sufficient and necessary conditions for multiple Alexandrov spaces being glued to an Alexandrov space. We propose a Gluing Conjecture, which says that the finite gluing of Alexandrov spaces is an Alexandrov…
A classical Theorem of Alexandrov states that the map associating its boundary to a convex polyhdedron of the 3-dimensional Euclidean space is a bijection from the set of convex polyhdedron up to congruence to the set of isometry classes of…
We present an algorithm that enumerates and classifies all edge-to-edge gluings of unit squares that correspond to convex polyhedra. We show that the number of such gluings of $n$ squares is polynomial in $n$, and the algorithm runs in time…
We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex…
We investigate the folding problem that asks if a polygon P can be folded to a polyhedron Q for given P and Q. Recently, an efficient algorithm for this problem has been developed when Q is a box. We extend this idea to regular polyhedra,…
Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In…
This paper is devoted to prove that if an Alexandrov space of curvature not less than $\kappa$ with a codimension one extremal subset which admits an isometric involution with respect to the induced length metric, then the metric space…
A polyhedron $\textbf{P} \subset \mathbb{R}^3$ has Rupert's property if a hole can be cut into it, such that a copy of $\textbf{P}$ can pass through this hole. There are several works investigating this property for some specific polyhedra:…
The rigidity theorems of Alexandrov (1950) and Stoker (1968) are classical results in the theory of convex polyhedra. In this paper we prove analogues of them for normal (resp., standard) ball-polyhedra. Here, a ball-polyhedron means an…
We show that a realization of a closed connected PL-manifold of dimension n-1 in n-dimensional Euclidean space (n>2) is the boundary of a convex polyhedron (finite or infinite) if and only if the interior of each (n-3)-face has a point,…
We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts. By contrast, the best previous such result used exponentially many cuts. More precisely,…
We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with $d\ge 3$ edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are…
An edge-unfolding of a polyhedron is produced by cutting along edges and flattening the faces to a *net*, a connected planar piece with no overlaps. A *grid unfolding* allows additional cuts along grid edges induced by coordinate planes…
We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\colon X=\amalg X_\ell\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the…
We consider the problem of determining the length of the shortest paths between points on the surfaces of tetrahedra and cubes. Our approach parallels the concept of Alexandrov's star unfolding but focuses on specific polyhedra and uses…
We propose an algorithm for deciding whether a given braid is pseudo-Anosov, reducible, or periodic. The algorithm is based on Garside's weighted decomposition and is polynomial-time in the word-length of an input braid. Moreover, a…
The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by…
We prove a gluing theorem which allows to construct an ample divisor on a rational surface from two given ample divisors on simpler surfaces. This theorem combined with the Cremona action on the ample cone gives rise to an algorithm for…
We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration…