Related papers: Highly neighborly centrally symmetric spheres

Nevo, Santos, and Wilson constructed $2^{\Omega(N^d)}$ combinatorially distinct simplicial $(2d-1)$-spheres with $N$ vertices. We prove that all spheres produced by one of their methods are shellable. Combining this with prior results of…

Zauner's conjecture concerns the existence of $d^2$ equiangular lines in $\mathbb{C}^d$; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs…

Stanley proved that for any centrally symmetric simplicial $d$-polytope $P$ with $d\geq 3$, $g_2(P) \geq {d \choose 2}-d$. We provide a characterization of centrally symmetric $d$-polytopes with $d\geq 4$ that satisfy this inequality as…

We construct non-constructible simplicial $d$-spheres with $d+10$ vertices and non-constructible, non-realizable simplicial $d$-balls with $d+9$ vertices for $d\geq 3$.

Smooth axially symmetric Helfrich topological spheres are either round or else they must satisfy a second order equation known as the reduced membrane equation [17]. In this paper, we show that, conversely, axially symmetric closed genus…

This article is covered by the article arxiv.1012.0925 We study intersection of two polyhedral spheres without self-intersections in 3-space. We find necessary and sufficient conditions on sequences x = x_1,x_2,...,x_n, y = y_1,y_2,...,y_n…

For fixed $d\geq 3$, we construct subsets of the $d$-dimensional lattice cube $[n]^d$ of size $n^{\frac{3}{d + 1} - o(1)}$ with no $d+2$ points on a sphere or a hyperplane. This improves the previously best known bound of…

We study intersection of two polyhedral spheres without self-intersections in 3-space. We find necessary and sufficient conditions on sequences x = x_1,x_2,...,x_n, y = y_1,y_2,...,y_n of positive integers, for existence of 2-dimensional…

We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes…

The Hirsch conjecture, posed in 1957, stated that the graph of a $d$-dimensional polytope or polyhedron with $n$ facets cannot have diameter greater than $n - d$. The conjecture itself has been disproved, but what we know about the…

Choice of an appropriate (3+1)-foliation of spacetime or a (2+1)-foliation of the Cauchy space, leads often to a substantial simplification of various mathematical problems in General Relativity Theory. We propose a new method to construct…

Motivated by intuitions from projective algebraic geometry, we provide a novel construction of subsets of the $d$-dimensional grid $[n]^d$ of size $n - o(n)$ with no $d + 2$ points on a sphere or a hyperplane. For $d = 2$, this improves the…

For positive integers $n,d$, let $\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere which admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex. We show that…

In this paper we investigate the simplicial structure of a chain complex associated to the higher order Hochschild homology over the $3$-sphere. We also introduce the tertiary Hochschild homology corresponding to a quintuple…

We present explicit constructions of centrally symmetric polytopes with many faces: first, we construct a d-dimensional centrally symmetric polytope P with about (1.316)^d vertices such that every pair of non-antipodal vertices of P spans…

For each $d\geq 3$ we construct cube complexes homeomorphic to the $d$-sphere with $n$ vertices in which the number of facets (assuming $d$ constant) is $\Omega(n^{5/4})$. This disproves a conjecture of Kalai's stating that the number of…

Spherical t-designs are Chebyshev-type averaging sets on the d-sphere S^d which are exact for polynomials of degree at most t. This concept was introduced in 1977 by Delsarte, Goethals, and Seidel, who also found the minimum possible size…

Working in univalent foundations, we investigate the symmetries of spheres, i.e., the types of the form $\mathbb{S}^n = \mathbb{S}^n$. The case of the circle has a slick answer: the symmetries of the circle form two copies of the circle.…

A combinatorial theorem on families of disjoint sub-boxes of a discrete cube, which implies that there at most $2^{d+1}-2$ neighbourly simplices in $\mathbb R^d$, is presented.

For integers $d \geq 2$ and $\epsilon = 0$ or 1, let $S^{1, d - 1}(\epsilon)$ denote the sphere product $S^{1} \times S^{d - 1}$ if $\epsilon = 0$ and the twisted $S^{d - 1}$ bundle over $S^{1}$ if $\epsilon = 1$. The main results of this…