Minimum Convex Partitions and Maximum Empty Polytopes

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.Comment: 16 pages, 4 figures; revised write-up with some running times improve

On the Beer index of convexity and its variants

Let $S$ be a subset of $\mathbb{R}^d$ with finite positive Lebesgue measure. The Beer index of convexity $\operatorname{b}(S)$ of $S$ is the probability that two points of $S$ chosen uniformly independently at random see each other in $S$. The convexity ratio $\operatorname{c}(S)$ of $S$ is the Lebesgue measure of the largest convex subset of $S$ divided by the Lebesgue measure of $S$. We investigate the relationship between these two natural measures of convexity. We show that every set $S\subseteq\mathbb{R}^2$ with simply connected components satisfies $\operatorname{b}(S)\leq\alpha\operatorname{c}(S)$ for an absolute constant $\alpha$, provided $\operatorname{b}(S)$ is defined. This implies an affirmative answer to the conjecture of Cabello et al. that this estimate holds for simple polygons. We also consider higher-order generalizations of $\operatorname{b}(S)$. For $1\leq k\leq d$, the $k$-index of convexity $\operatorname{b}_k(S)$ of a set $S\subseteq\mathbb{R}^d$ is the probability that the convex hull of a $(k+1)$-tuple of points chosen uniformly independently at random from $S$ is contained in $S$. We show that for every $d\geq 2$ there is a constant $\beta(d)>0$ such that every set $S\subseteq\mathbb{R}^d$ satisfies $\operatorname{b}_d(S)\leq\beta\operatorname{c}(S)$, provided $\operatorname{b}_d(S)$ exists. We provide an almost matching lower bound by showing that there is a constant $\gamma(d)>0$ such that for every $\varepsilon\in(0,1)$ there is a set $S\subseteq\mathbb{R}^d$ of Lebesgue measure $1$ satisfying $\operatorname{c}(S)\leq\varepsilon$ and $\operatorname{b}_d(S)\geq\gamma\frac{\varepsilon}{\log_2{1/\varepsilon}}\geq\gamma\frac{\operatorname{c}(S)}{\log_2{1/\operatorname{c}(S)}}$.Comment: Final version, minor revisio

Piercing Quasi-Rectangles: On a Problem of Danzer and Rogers

It is an old problem of Danzer and Rogers to decide whether it is possible arrange O(1/epsilon) points in the unit square so that every rectangle of area epsilon contains at least one of them. We show that the answer to this question is in the negative if we slightly relax the notion of rectangles, as follows. Let delta be a fixed small positive number. A quasi-rectangle is a region swept out by a continuously moving segment s, with no rotation, so that throughout the motion the angle between the trajectory of the center of s and its normal vector remains at most delta. We show that the smallest number of points needed to pierce all quasi-rectangles of area e is Theta (1/epsilon log 1/epsilon)

Stable Isotopes in Tree Rings

This Open Access volume highlights how tree ring stable isotopes have been used to address a range of environmental issues from paleoclimatology to forest management, and anthropogenic impacts on forest growth. It will further evaluate weaknesses and strengths of isotope applications in tree rings. In contrast to older tree ring studies, which predominantly applied a pure statistical approach this book will focus on physiological mechanisms that influence isotopic signals and reflect environmental impacts. Focusing on connections between physiological responses and drivers of isotope variation will also clarify why environmental impacts are not linearly reflected in isotope ratios and tree ring widths. This volume will be of interest to any researcher and educator who uses tree rings (and other organic matter proxies) to reconstruct paleoclimate as well as to understand contemporary functional processes and anthropogenic influences on native ecosystems. The use of stable isotopes in biogeochemical studies has expanded greatly in recent years, making this volume a valuable resource to a growing and vibrant community of researchers