Related papers: The minimum angle condition for $d$-simplices
We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index $n$. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a…
Products of simplices, called simplotopes, and their triangulations arise naturally in algorithmic applications in game theory and optimization. We develop techniques to derive lower bounds for the size of simplicial covers and…
In this paper the problem of finding a normal form of triangles and plane quadrilaterals up to similarity is considered. Several normal forms for triangles and a normal form for quadrilaterals of special case are described. Normal forms of…
Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area…
We study the regularity of minimizers to the composite membrane problem in the plane (ie given a domain omega and a positive number A, smaller than the measure of omega, minimize the first Dirichlet eigenvalue for the Schrodinger operator…
A characterization of simplicial objects in categories with finite products obtained by the reduced bar construction is given. The condition that characterizes such simplicial objects is a strictification of Segal's condition guaranteeing…
Attending to an open problem in the literature stated by Mirzakhani and Vondr\'ak, we give a lower bound of the number of non-monochromatic simplices for Sperner labelings of the vertices of a triangulation of a given $ k$-simplex with…
This paper introduces three sets of sufficient conditions, for generating bijective simplicial mappings of manifold meshes. A necessary condition for a simplicial mapping of a mesh to be injective is that it either maintains the orientation…
We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H…
In this article we will consider average angles of triangle, which share the same side with regular polygons. In particular we will count average angles in the triangle, which share the same bottom side with a square with length side $d=1$.
We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general…
We prove hyperbolicity of global minimizers for random Lagrangian systems in dimension 1. The proof considerably simplifies a related result in [2]. The conditions for hyperbolicity are almost optimal: they are essentially the same as…
In answering questions from arXiv:0901.2337v1 we prove a triangulation result that is of independent interest. In more detail, let R be an o-minimal field with a proper convex subring V, and let st: V \to k be the corresponding standard…
We present a short proof, relaying on the divergence theorem, verifying that minimal sets in the plane are trivial.
An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These…
Some sufficient conditions on a simplicial space $X$ guaranteeing that $X_1\simeq \Omega|X|$ were given by Segal. We give a generalization of this result for multisimplicial spaces. This generalization is appropriate for the reduced bar…
In [5] I solved the Thom's conjecture that a proper Thom map is triangulable. In this paper I drop the properness condition in the semialgebraic case and, moreover, in the definable case in an o-minimal structure.
We consider compact connected minimal surfaces, with a pair of boundary curves (not necessarily convex) in distinct planes, that have least-area amongst all orientable surfaces with the same boundary. When the planes containing these two…
A completely well-centered tetrahedral mesh is a triangulation of a three dimensional domain in which every tetrahedron and every triangle contains its circumcenter in its interior. Such meshes have applications in scientific computing and…
A simplicial polytope is a polytope with all its facets being combinatorially equivalent to simplices. We deal with the edge connectivity of the graphs of simplicial polytopes. We first establish that, for any $d\ge 3$, for any $d\ge 3$,…