Related papers: A rigidity criterion for non-convex polyhedra

With respect to earlier investigations, the theory of multi-component, concentric, copolar, axisymmetric, rigidly rotating polytropes is improved and extended, including subsystems with nonzero density on the boundary and subsystems with…

A polytope is called indecomposable if it cannot be expressed nontrivially as a Minkowski sum of other polytopes. Since Gale introduced the concept in 1954, several increasingly strong criteria have been developed to characterize…

It is proved that any smooth manifold $\mathcal M$ of dimension $m$ admits a smooth polynomially convex embedding into $\mathbb C^n$ when $n\geq \lfloor 5m/4\rfloor$. Further, such embeddings are dense in the space of smooth maps from…

We show that for almost every $(P,\lambda)$ where $P$ is a convex polygon and $\lambda\in(0,1)$, the corresponding outer billiard about $P$ with contraction $\lambda$ is asymptotically periodic, i.e., has a finite number of periodic orbits…

Polyhedra are generically rigid, but can be made to flex under certain symmetry conditions. We generalise Raoul Bricard's 1897 method for making flexible octahedra to construct an infinite family of flexible polyhedra with…

Nontrivial infinitesimal bendings for a class of two-dimensional surfaces are constructed. The surfaces considered here are orientable; compact; with boundary; have positive curvature everywhere except at finitely many planar points; and…

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a…

In this note we prove two ellipsoid characterization theorems. The first one is that if $K$ is a convex body in a normed space with unit ball $M$, and for any point $p \notin K$ and in any 2-dimensional plane $P$ intersecting $\inter K$ and…

In this paper, we prove that every real analytic totally nondegenerate model CR manifold of length >= 3 has rigidity. This result was actually conjectured before by Valerii Beloshapka as the so-called "maximum conjecture". It follows that…

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:…

Perez proved some $L^2$ inequalities for closed convex hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, more generally, for closed hypersurfaces with non-negative Ricci curvature, immersed in an Einstein manifold. In this…

The Stoker problem, first formulated in 1968, consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for…

We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed…

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…

Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes…

Let $P$ be a geodesic plane in a convex cocompact, acylindrical hyperbolic 3-manifold $M$. Assume that $P^*=M^*\cap P$ is nonempty, where $M^*$ is the interior of the convex core of $M$. Does this condition imply that $P$ is either closed…

This work provides two sufficient conditions in terms of sections or projections for a convex body to be a polytope. These conditions are necessary as well.

We give an example of an infinitesimally nonrigid polyhedron in the Lobachevsky 3-space and construct an infinitesimal flex of that polyhedron such that the volume of the polyhedron isn't stationary under the flex.

Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set are locally…

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.