Small Strong Epsilon Nets

Let P be a set of n points in $\mathbb{R}^d$. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than $dn\over d+1$ points of P. We call a point x a strong centerpoint for a family of objects $\mathcal{C}$ if $x \in P$ is contained in every object $C \in \mathcal{C}$ that contains more than a constant fraction of points of P. A strong centerpoint does not exist even for halfspaces in $\mathbb{R}^2$. We prove that a strong centerpoint exists for axis-parallel boxes in $\mathbb{R}^d$ and give exact bounds. We then extend this to small strong $\epsilon$-nets in the plane and prove upper and lower bounds for $\epsilon_i^\mathcal{S}$ where $\mathcal{S}$ is the family of axis-parallel rectangles, halfspaces and disks. Here $\epsilon_i^\mathcal{S}$ represents the smallest real number in $[0,1]$ such that there exists an $\epsilon_i^\mathcal{S}$-net of size i with respect to $\mathcal{S}$.Comment: 19 pages, 12 figure

On Strong Centerpoints

Let $P$ be a set of $n$ points in $\mathbb{R}^d$ and $\mathcal{F}$ be a family of geometric objects. We call a point $x \in P$ a strong centerpoint of $P$ w.r.t $\mathcal{F}$ if $x$ is contained in all $F \in \mathcal{F}$ that contains more than $cn$ points from $P$, where $c$ is a fixed constant. A strong centerpoint does not exist even when $\mathcal{F}$ is the family of halfspaces in the plane. We prove the existence of strong centerpoints with exact constants for convex polytopes defined by a fixed set of orientations. We also prove the existence of strong centerpoints for abstract set systems with bounded intersection

On the mathematical and foundational significance of the uncountable

We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindel\"of lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindel\"of property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. $[0,1]$ as 'almost finite', while the latter allows one to treat uncountable sets like e.g. $\mathbb{R}$ as 'almost countable'. This reduction of the uncountable to the finite/countable turns out to have a considerable logical and computational cost: we show that the aforementioned lemmas, and many related theorems, are extremely hard to prove, while the associated sub-covers are extremely hard to compute. Indeed, in terms of the standard scale (based on comprehension axioms), a proof of these lemmas requires at least the full extent of second-order arithmetic, a system originating from Hilbert-Bernays' Grundlagen der Mathematik. This observation has far-reaching implications for the Grundlagen's spiritual successor, the program of Reverse Mathematics, and the associated G\"odel hierachy. We also show that the Cousin lemma is essential for the development of the gauge integral, a generalisation of the Lebesgue and improper Riemann integrals that also uniquely provides a direct formalisation of Feynman's path integral.Comment: 35 pages with one figure. The content of this version extends the published version in that Sections 3.3.4 and 3.4 below are new. Small corrections/additions have also been made to reflect new development

Selection Lemmas for various geometric objects

Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set. In the first selection lemma, we consider the set of all the objects induced (spanned) by a point set $P$. This question has been widely explored for simplices in $\mathbb{R}^d$, with tight bounds in $\mathbb{R}^2$. In our paper, we prove first selection lemma for other classes of geometric objects. We also consider the strong variant of this problem where we add the constraint that the piercing point comes from $P$. We prove an exact result on the strong and the weak variant of the first selection lemma for axis-parallel rectangles, special subclasses of axis-parallel rectangles like quadrants and slabs, disks (for centrally symmetric point sets). We also show non-trivial bounds on the first selection lemma for axis-parallel boxes and hyperspheres in $\mathbb{R}^d$. In the second selection lemma, we consider an arbitrary $m$ sized subset of the set of all objects induced by $P$. We study this problem for axis-parallel rectangles and show that there exists an point in the plane that is contained in $\frac{m^3}{24n^4}$ rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir when $m$ is almost quadratic

Globalization and the Role of Public Transfers in Redistributing Income in Latin America and the Caribbean

This paper focuses on measuring the extent to which publicly subsidized transfers in Latin America and the Caribbean redistribute income. The redi$social protection, insurance, redistribution, targeting, poverty, inequality, welfare