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

Stabbing Convex Bodies with Lines and Flats

$\newcommand{\eps}{\varepsilon}\newcommand{\tldO}{\widetilde{O}}$Consider the problem of constructing weak \eps-nets where the stabbing elements are lines or $k$-flats instead of points. We study this problem in the simplest setting where it is still interesting -- namely, the uniform measure of volume over the hypercube $[0,1]^d\bigr.$. Specifically, a (k,\eps)-net is a set of $k$-flats, such that any convex body in $[0,1]^d$ of volume larger than \eps is stabbed by one of these $k$-flats. We show that for $k \geq 1$, one can construct (k,\eps)-nets of size O(1/\eps^{1-k/d}). We also prove that any such net must have size at least \Omega(1/\eps^{1-k/d}). As a concrete example, in three dimensions all \eps-heavy bodies in $[0,1]^3$ can be stabbed by \Theta(1/\eps^{2/3}) lines. Note, that these bounds are \emph{sublinear} in 1/\eps, and are thus somewhat surprising. The new construction also works for points providing a weak \eps-net of size O(\tfrac{1}{\eps}\log^{d-1} \tfrac{1}{\eps} ).Comment: 13 pages, 6 figures; updated with improved constructions of $(k, \varepsilon)$-nets for $k \geq 1

New Constructions of Weak ε-Nets

A finite set $N \subset \R^d$ is a {\em weak \eps-net} for an $n$-point set $X\subset \R^d$ (with respect to convex sets) if $N$ intersects every convex set $K$ with |K\,\cap\,X|\geq \eps n. We give an alternative, and arguably simpler, proof of the fact, first shown by Chazelle et al., that every point set $X$ in $\R^d$ admits a weak \eps-net of cardinality O(\eps^{-d}\polylog(1/\eps)). Moreover, for a number of special point sets (e.g., for points on the moment curve), our method gives substantially better bounds. The construction yields an algorithm to construct such weak \eps-nets in time $O(n\ln(1/\eps))

Journey to the Center of the Point Set

We revisit an algorithm of Clarkson et al. [K. L. Clarkson et al., 1996], that computes (roughly) a 1/(4d^2)-centerpoint in O~(d^9) time, for a point set in R^d, where O~ hides polylogarithmic terms. We present an improved algorithm that computes (roughly) a 1/d^2-centerpoint with running time O~(d^7). While the improvements are (arguably) mild, it is the first progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm