A Danzer set for Axis Parallel Boxes

We present concrete constructions of discrete sets in $\mathbb{R}^d$ ($d\ge 2$) that intersect every aligned box of volume $1$ in $\mathbb{R}^d$, and which have optimal growth rate $O(T^d)$

Dense forests and Danzer sets

A set $Y\subseteq\mathbb{R}^d$ that intersects every convex set of volume $1$ is called a Danzer set. It is not known whether there are Danzer sets in $\mathbb{R}^d$ with growth rate $O(T^d)$. We prove that natural candidates, such as discrete sets that arise from substitutions and from cut-and-project constructions, are not Danzer sets. For cut and project sets our proof relies on the dynamics of homogeneous flows. We consider a weakening of the Danzer problem, the existence of uniformly discrete dense forests, and we use homogeneous dynamics (in particular Ratner's theorems on unipotent flows) to construct such sets. We also prove an equivalence between the above problem and a well-known combinatorial problem, and deduce the existence of Danzer sets with growth rate $O(T^d\log T)$, improving the previous bound of $O(T^d\log^{d-1} T)$

Helly-Type Theorems in Property Testing

Helly's theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If $S$ is a set of $n$ points in $R^d$, we say that $S$ is $(k,G)$-clusterable if it can be partitioned into $k$ clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object $G$. In this paper, as an application of Helly's theorem, by taking a constant size sample from $S$, we present a testing algorithm for $(k,G)$-clustering, i.e., to distinguish between two cases: when $S$ is $(k,G)$-clusterable, and when it is $\epsilon$-far from being $(k,G)$-clusterable. A set $S$ is $\epsilon$-far $(0<\epsilon\leq1)$ from being $(k,G)$-clusterable if at least $\epsilon n$ points need to be removed from $S$ to make it $(k,G)$-clusterable. We solve this problem for $k=1$ and when $G$ is a symmetric convex object. For $k>1$, we solve a weaker version of this problem. Finally, as an application of our testing result, in clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability

Coloring translates and homothets of a convex body

We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in \RR^n.Comment: 11 pages, 2 figure

Stabbing boxes with finitely many axis-parallel lines and flats

We give necessary and sufficient condition for an infinite collection of axis-parallel boxes in $\mathbb{R}^{d}$ to be pierceable by finitely many axis-parallel $k$-flats, where $0 \leq k < d$. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner $(p,q)$-problem.Comment: 13 page

Geometric Permutations of Non-Overlapping Unit Balls Revisited

Given four congruent balls $A, B, C, D$ in $R^{d}$ that have disjoint interior and admit a line that intersects them in the order $ABCD$, we show that the distance between the centers of consecutive balls is smaller than the distance between the centers of $A$ and $D$. This allows us to give a new short proof that $n$ interior-disjoint congruent balls admit at most three geometric permutations, two if $n\ge 7$. We also make a conjecture that would imply that $n\geq 4$ such balls admit at most two geometric permutations, and show that if the conjecture is false, then there is a counter-example of a highly degenerate nature