Related papers: Hadwiger and Helly-type theorems for disjoint unit…

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 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 study a family of self-affine tiles in $\mathbb{R}^d$ ($d\ge2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Deng and Lau in $\mathbb{R}^2$ and its extension to $\mathbb{R}^3}$ by the authors.…

We show that the family of $m$-dimensional isotropic projections in $\R^{2n}$ is transversal. As an application we show that the Besicovitch-Federer projection theorem holds for isotropic projections. We also use transversality to obtain…

This work is a continuation of [13]. We study the linear disjointness between higher-order oscillating sequences and nonlinear dynamical systems. Specifically, we prove that any oscillating sequence of order $m=d+k-1$ and any simple…

We complete the classification of the smooth, closed, oriented 4-manifolds having Euler characteristic less than four and a horizontal handlebody decomposition of genus one. We use the classification result to find a large family of…

We show the optimal coherence of $2d$ lines in $\mathbb{C}^{d}$ is given by the Welch bound whenever a skew Hadamard of order $d+1$ exists. Our proof uses a variant of Hadamard doubling that converts any equiangular tight frame of size…

A celebrated theorem of Kleitman in extremal combinatorics states that a collection of binary vectors in $\{0, 1\}^n$ with diameter $d$ has cardinality at most that of a Hamming ball of radius $d/2$. In this paper, we give an algebraic…

Two celebrated extensions of Helly's theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of B\'ar\'any, Katchalski, and Pach (1982). Improving on several recent works, we prove an…

We prove an extension of a ham sandwich theorem for families of lines in the plane by Dujmovi\'{c} and Langerman. Given two sets $A, B$ of $n$ lines each in the plane, we prove that it is possible to partition the plane into $r$ convex…

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following…

Research on Helly-type theorems in combinatorial convex geometry has produced volumetric versions of Helly's theorem using witness sets and quantitative extensions of Doignon's theorem. This paper combines these philosophies and presents…

We give necessary and sufficient condition so that we have d-hypercyclicity for operators who map a holomorphic function to a partial sum of the Taylor expansion. This problem is connected with doubly universal Taylors series and this is an…

In this paper, we consider the embedding of a complete $d$-uniform geometric hypergraph with $n$ vertices in general position in $\mathbb{R}^d$, where each hyperedge is represented as a $(d-1)$-simplex, and a pair of hyperedges is defined…

We study parameters of the convexity spaces associated with families of sets in $\mathbb{R}^d$ where every intersection between $t$ sets of the family has its Betti numbers bounded from above by a function of $t$. Although the Radon number…

The necessary and sufficient conditions under which a given family $\mathcal{F}$ of subsets of finite set $X$ coincides with the family $\mathbf{B}_X$ of all balls generated by some ultrametric $d$ on $X$ are found. It is shown that the…

Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either divisible by $n$ or by a…

In this paper, we study the $d$-dimensional rectilinear drawings of the complete $d$-uniform hypergraph $K_{2d}^d$. Anshu et al. [Computational Geometry: Theory and Applications, 2017] used Gale transform and Ham-Sandwich theorem to prove…

We prove a new fractional Helly theorem for families of sets obeying topological conditions. More precisely, we show that the nerve of a finite family of open sets (and of subcomplexes of cell complexes) in R^d is k-Leray where k depends on…

We prove a conjecture of D. Oberlin on the dimension of unions of lines in $\mathbb{R}^n$. If $d \geq 1$ is an integer, $0 \leq \beta \leq 1$, and $L$ is a set of lines in $\mathbb{R}^n$ with Hausdorff dimension at least $2(d-1) + \beta$,…