Related papers: Edge-Unfolding Nearly Flat Convex Caps

This paper deals with finding surfaces in $\mathbb{R}^3$ which are as close as possible to being flat and span a given contour such that the contour is a geodesic on the sought surface. We look for a surface which minimizes the total…

A convex geometric graph is a graph whose vertices are the corners of a convex polygon P in the plane and whose edges are boundary edges and diagonals of the polygon. It is called triangulation-free if its non-boundary edges do not contain…

Approximating convex bodies is a fundamental problem in geometry. Given a convex body $K$ in $\mathbb{R}^d$ for a fixed dimension $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff…

In this note we show that unbounded convex polygons with nonparallel unbounded edges are polynomial images of ${\mathbb R}^2$, whereas their interiors are polynomial images of ${\mathbb R}^3$

We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than…

Let $(M,g)$ be an asymptotically flat $3$-manifold containing no closed embedded minimal surfaces. We prove that for every point $p\in M$ there exists a complete properly embedded minimal plane in $M$ containing $p$.

We give a simple proof of the following result: There exists a non-convex polyhedron whose surface is isometric to the surface of a cube of smaller volume.

We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in…

In the paper there are described new examples of conformally flat three dimensional almost cosymplectic manifolds. All these manifolds form a class which was completely characterized.

The aim of this paper is to give a survey of the known results concerning centrally symmetric polytopes, spheres, and manifolds. We further enumerate nearly neighborly centrally symmetric spheres and centrally symmetric products of spheres…

A particular Riemannian metric which originally has been obtained for a well-known coordinate system in the Euclidean 3-space, is shown to specify, in fact, a manifold with boundary. There are two ways to make the manifold complete. One is…

Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', $m$. Such polygons are called \emph{$m$-convex} polygons and are characterised by…

An n-gon is defined as a sequence \P=(V_0,...,V_{n-1}) of n points on the plane. An n-gon \P is said to be convex if the boundary of the convex hull of the set {V_0,...,V_{n-1}} of the vertices of \P coincides with the union of the edges…

We show that any compact half-conformally flat manifold of negative type, with bounded $L^2$ energy, sufficiently small scalar curvature, and a non-collapsing assumption, has all betti numbers bounded. We show that this result is optimal…

Two single parameter families of polyhedra $P(\psi)$ are constructed in three dimensional spaces of constant curvature $C(\psi)$. Identification of the faces of the polyhedra via isometries results in cone manifolds $M(\psi)$ which are…

We prove that the image of an isometric embedding into ${\mathbb R}^3$ of a two dimensionnal complete Riemannian manifold $(\Sigma, g)$ without boundary is a convex surface provided both the embedding and the metric $g$ enjoy a…

A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$ f((x+y)/2) \le (f(x)+f(y))/2 + 1. $$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper…

In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…

We show that every ridge unfolding of an $n$-cube is without self-overlap, yielding a valid net. The results are obtained by developing machinery that translates cube unfolding into combinatorial frameworks. Moreover, the geometry of the…

In this paper, we study the deformation of the n-dimensional strictly convex hypersurface in $\mathbb R^{n+1}$ whose speed at a point on the hypersurface is proportional to $\alpha$-power of positive part of Gauss Curvature. For…