相关论文: Maximal triangulations of a regular prism
We consider a spherical antiprism. It is a convex polyhedron with $2n$ vertices in the spherical space $\mathbb{S}^3$. This polyhedron has a group of symmetries $S_{2n}$ generated by a mirror-rotational symmetry of order $2n$, i.e. rotation…
We prove that triangulations with maximum degree at most 5 satisfy the List-Edge-Coloring Conjecture.
It is known that any infinite frieze comes from a triangulation of an annulus by Baur, Parsons and Tschabold. In this paper we show that each periodic infinite frieze determines a triangulation of an annulus in essentially a unique way.…
A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in $\mathbb{R}^n$, using semidefinite programming…
In this paper, we show that the maximum number of points in $d\geq3$ dimensions determining exactly 2 distinct triangles is $2d$. We further show that this maximum is uniquely achieved by the vertices of the $d$-orthoplex. We build upon the…
We consider right prisms with horizontal quadrilateral bases and tops, and vertical rectangular sides. We look for examples where all the edges, face diagonals and space diagonals are integers. We find examples when the base is an isosceles…
Let $h(n)$ denote the maximum number of triangles with angles between $59^\circ$ and $61^\circ$ in any $n$-element planar set. Our main result is an exact formula for $h(n)$. We also prove $h(n)= n^3/24+ O(n \log n)$ as $n\to \infty$.…
We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically $12^n…
We utilize the obstruction theory of Galewski-Matumoto-Stern to derive equivalent formulations of the Triangulation Conjecture. For example, every closed topological manifold M^n with n > 4 can be simplicially triangulated if and only if…
We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with $n$ vertices. We give exact algorithms that solve these problems in time…
We study maximal regularity in interpolation spaces for the sum of three closed linear operators on a Banach space, and we apply the abstract results to obtain Besov and H\"older maximal regularity for complete second order Cauchy problems…
We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an…
In this paper, we obtain the counting formulaes of convex pentagons and convex hexagons, respectively, in an $n$-triangular net by solving the corresponding recursive formulaes.
We propose a conjecture for the limit of mean-field spin glasses with a bipartite structure, and show that the conjectured limit is an upper bound. The conjectured limit is described in terms of the solution of an infinite-dimensional…
Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of…
P. Arnoux and A. Marin showed that any triangulation of $\mathbb{RP}^n$ contains more than $\frac{(n+1)(n+2)}{2}$ vertices if $n \geq 3$. We construct some natural triangulation of $\mathbb{RP}^n$ with $\frac{n(n+5)}{2}-1$ vertices for all…
The aim of this paper is the determination of the largest $n$-dimensional polytope with $n+3$ vertices of unit diameter. This is a special case of a more general problem proposed by Graham.
In this paper, claims by Lemmens and Seidel in 1973 about equiangular sets of lines with angle $1/5$ are proved by carefully analyzing pillar decompositions, with the aid of the uniqueness of two-graphs on $276$ vertices. The Neumann…
The antiprism triangulation provides a natural way to subdivide a simplicial complex $\Delta$, similar to barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as…
The objects of study are triangulations of the dilated standard triangle in the plane. Motivated by work on T-curves (Geiselmann et al., 2026), the focus lies on unimodular triangulations with a fixed symmetry axis. Lower and upper bounds…