Related papers: Hunting for reduced polytopes
In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only.…
We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results of Lann\'er, Kaplinskaja,…
Given a set $S \subseteq \mathbb{R}^d$, a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum number of vertices of hollow polytopes whose facets are…
There are significant differences between Helmholtz and Hodge's decomposition theorems, but both share a common flavor. This paper is a first step to bring them together. We here produce Helmholtz theorems for differential 1-forms and…
This paper gives, in generic situations, a complete classification of ruled minimal surfaces in pseudo-Euclidean space with arbitrary index. In addition, we discuss the condition for ruled minimal surfaces to exist, and give a…
A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete…
We prove global existence of solutions to the Cauchy problem for the compressible Navier-Stokes equations in Euclidean spaces, given initial data with small norms in Besov and critical weighted Besov spaces. Global existence and a priori…
A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher…
We have developed a general method for finding apparent horizons in 3D numerical relativity. Instead of solving for the partial differential equation describing the location of the apparent horizons, we expand the closed 2D surfaces in…
A lower bound for the number of 3-periodical billiard trajectories in a manifold embedded in Euclidean space is obtained.
Main goal of this paper is to generalize Hadamard's real part theorem and invariant forms of Borel-Carath\'eodory's theorem from complex analysis to solutions of the Riesz system in the three-dimensional Euclidean space in the framework of…
We show that: (1) unimodular simplices in a lattice 3-polytope cover a neighborhood of the boundary of the polytope if and only if the polytope is very ample, (2) the convex hull of lattice points in every ellipsoid in R^3 has a unimodular…
We establish some characterizations of elliptic hyperboloids (resp., ellipsoids) in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$, using the $n$-dimensional area of the sections cut off by hyperplanes and the $(n+1)$-dimensional…
The paper concerns the theory of parabolic equations on a broad class of closed subsets of Euclidean space possessing a kind of tangent structure. A necessary framework for considering evolutionary problems is developed, and fundamental…
A shape of a combinatorial polytope is a convex embedding into Euclidean space. We provide necessary and sufficient conditions for a piecewise linear map between two shapes of the same polytope to be a compression (respectively a weak…
2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level…
A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of $P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$ vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective…
The Continuous p-Dispersion Problem (CpDP) with boundary constraints asks for the placement of a fixed number of points in a compact subset of Euclidean space such that the minimum distance between any two points, as well as the points and…
We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$, $m\leq n$, we define and prove the existence of a partition of…
A subset of a convex body $B$ containing the origin in a Euclidean space is {\it parkable in $B$} if it can be translated inside $B$ in such a manner that the translate the origin. We provide characterizations of ellipsoids and of centrally…