Related papers: No-dimensional Tverberg Partitions Revisited

Let $P$ be a set of $n$ points in the plane. We compute the value of $\theta\in [0,2\pi)$ for which the rectilinear convex hull of $P$, denoted by $\mathcal{RH}_\theta(P)$, has minimum (or maximum) area in optimal $O(n\log n)$ time and…

For a planar point set $P$, its convex hull is the smallest convex polygon that encloses all points in $P$. The construction of the convex hull from an array $I_P$ containing $P$ is a fundamental problem in computational geometry. By…

Let $(P,E)$ be a $(d+1)$-uniform geometric hypergraph, where $P$ is an $n$-point set in general position in $\mathbb{R}^d$ and $E\subseteq {P\choose d+1}$ is a collection of $\epsilon{n\choose d+1}$ $d$-dimensional simplices with vertices…

Let $\Gamma$ be a $Q$-polynomial distance-regular graph with diameter at least $3$. Terwilliger (1993) implicitly showed that there exists a polynomial, say $T(\lambda)\in \mathbb{C}[\lambda]$, of degree $4$ depending only on the…

$\newcommand{\Re}{\mathbb{R}}$We study the minWSPD problem of computing the minimum-size well-separated pairs decomposition of a set of points, and show constant approximation algorithms in low-dimensional Euclidean space and doubling…

Given $n$ non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into $O(n^{3/2}\mathop{\mathrm{polylog}} n)$ pieces, such that the depth relation among these pieces…

Let $S = \{ {A_1},{A_2}, \cdots ,{A_n}\} $ be a finite point set in m-dimensional Euclidean space ${E^m}$, and$\left\| {{A_i}{A_j}} \right\|$ be the distance between $A_i$ and $A_j$. Define $\sigma (S) = \sum\limits_{1 \le i < j \le n}…

One can associate to any bivariate polynomial P(X,Y) its Newton polygon. This is the convex hull of the points (i,j) such that the monomial X^i Y^j appears in P with a nonzero coefficient. We conjecture that when P is expressed as a sum of…

Given any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space ${\mathcal P}(M)$ of probability measures on $M$ that is (i) reversible w.r.t.~the entropic measure ${\mathbb P}^\beta$ on ${\mathcal P}(M)$,…

Let $\mathcal{T}$ be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles,…

Let $\D$ be a set of $n$ pairwise disjoint unit balls in $\R^d$ and $P$ the set of their center points. A hyperplane $\Hy$ is an \emph{$m$-separator} for $\D$ if each closed halfspace bounded by $\Hy$ contains at least $m$ points from $P$.…

Given two distinct point sets $P$ and $Q$ in the plane, we say that $Q$ \emph{blocks} $P$ if no two points of $P$ are adjacent in any Delaunay triangulation of $P\cup Q$. Aichholzer et al. (2013) showed that any set $P$ of $n$ points in…

A recent preprint of S. Kojima and G. McShane [KM] observes a beautiful explicit connection between Teichm\"uller translation distance and hyperbolic volume. It relies on a key estimate which we supply here: using geometric inflexibility of…

This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the…

We propose a geometric structure induced by any given convex polygon $P$, called $Nest(P)$, which is an arrangement of $\Theta(n^2)$ line segments, each of which is parallel to an edge of $P$, where $n$ denotes the number of edges of $P$.…

We prove the density of polyhedral partitions in the set of finite Caccioppoli partitions. Precisely, we consider a decomposition $u$ of a bounded Lipschitz set $\Omega\subset\mathbb R^n$ into finitely many subsets of finite perimeter,…

In Kirchheim, M\"{u}ller and \v{S}ver\'{a}k [Studying nonlinear PDE by geometry in matrix space. Geometric analysis and nonlinear partial differential equations, 2003], the authors proposed the program to use the differential inclusion…

A partition $\mathcal{P}$ of $\mathbb{R}^d$ is called a $(k,\varepsilon)$-secluded partition if, for every $\vec{p} \in \mathbb{R}^d$, the ball $\overline{B}_{\infty}(\varepsilon, \vec{p})$ intersects at most $k$ members of $\mathcal{P}$. A…

For a given $\lambda >0$, a convex body in $\mathbb R^n$ is $\lambda$-convex if it is the intersection of (finitely or infinitely many) balls of radius $1/\lambda$. In this note, we show that among all $\lambda$-convex bodies in $\mathbb…

A convex partition of a point set P in the plane is a planar partition of the convex hull of P with empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the…