Related papers: A note on the no-$(d+2)$-on-a-sphere problem

In a recent preprint on arXiv Roland Bacher showed that the number $p_d$ of non-similar perfect $d$-dimensional quadratic forms satisfies $e^{\Omega(d)} < p_d < e^{O(d^3\log(d))}$. We improve the upper bound to $e^{O(d^2\log(d))}$ by a…

We show there exists a packing of identical spheres in $\mathbb{R}^d$ with density at least \[ (1-o(1))\frac{d \log d}{2^{d+1}}\, , \] as $d\to\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is…

A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes passing below…

Pach showed that every $d+1$ sets of points $Q_1,\dotsc,Q_{d+1} \subset \mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. We show, by means of an example, that a…

We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced…

We show that if an open set in $\mathbb{R}^d$ can be fibered by unit $n$-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{ 0, 1, 3, 7 \right\}$. For these values of $n$, we…

We improve our earlier upper bound on the numbers of antipodal pairs of points among $n$ points in ${\mathbb{R}}^3$, to $2n^2/5+O(n^c)$, for some $c<2$. We prove that the minimal number of antipodal pairs among $n$ points in convex position…

Let $P$ be a set of $n$ points in real projective $d$-space, not all contained in a hyperplane, such that any $d$ points span a hyperplane. An ordinary hyperplane of $P$ is a hyperplane containing exactly $d$ points of $P$. We show that if…

We use polynomial method techniques to bound the number of tangent pairs in a collection of $N$ spheres in $\mathbb{R}^n$ subject to a non-degeneracy condition, for any $n \geq 3$. The condition, inspired by work of Zahl for $n=3$, asserts…

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower…

It was recently shown by Aka, Einsiedler and Shapira that if d>2, the set of primitive vectors on large spheres when projected to the d-1-dimensional sphere coupled with the shape of the lattice in their orthogonal complement equidistribute…

We have identified some necessary conditions for the existence of rigid sphere designs. In particular, we have successfully resolved the conjecture proposed by [Ban87]; Given fixed positive integers t and d, we show that there exist only…

In this paper, we prove the existence of nontrivial contractible domains $\Omega\subset\mathbb{S}^{d}$, $d\geq2$, such that the overdetermined elliptic problem \begin{equation*} \begin{cases} -\varepsilon\Delta_{g} u +u-u^{p}=0 &\mbox{in…

Let $d \geq 1$ and $s \leq 2^d$ be nonnegative integers. For a subset $A$ of vertices of the hypercube $Q_n$ and $n\geq d$, let $\lambda(n,d,s,A)$ denote the fraction of subcubes $Q_d$ of $Q_n$ that contain exactly $s$ vertices of $A$. Let…

We construct 2^{\Omega(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves…

The Heilbronn triangle problem asks for the placement of $n$ points in a unit square that maximizes the smallest area of a triangle formed by any three of those points. In $1972$, Schmidt considered a natural generalization of this problem.…

Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in $A$ if they are the vertices of a convex polytope which contains no points of $A$ in its interior. We construct arbitrarily large…

For special $d$-dimensional hyperbolic shells $E$ with $ d\geq 5$ we show that the number of lattice points in $E$ intersected with a $d$-dimensional cube $C_r$ of edge length $r$, can be approximated by the volume of $E\cap C_r$, as $r$…

We introduce the $k$-stellated spheres and compare and contrast them with $k$-stacked spheres. It is shown that for $d \geq 2k$, any $k$-stellated sphere of dimension $d$ bounds a unique and canonically defined $k$-stacked ball. In…

A small triangulation of the sphere product can be found in lower dimensions by computer search and is known in few other cases: Klee and Novik constructed a centrally symmetric triangulation of $\mathbb{S}^i\times \mathbb{S}^{d-i-1}$ with…