Discrete & Computational Geometry

List of Papers (Total 73)

Intrinsic Volumes of Polyhedral Cones: A Combinatorial Perspective

The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the general Steiner formula, the conic analogues of the Brianchon–Gram–Euler and the ...

Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings

An embedding $i \mapsto p_i\in \mathbb {R}^d$ of the vertices of a graph G is called universally completable if the following holds: For any other embedding $i\mapsto q_i~\in \mathbb {R}^{k}$ satisfying $q_i^{T}q_j = p_i^{T}p_j$ for $i = j$ and i adjacent to j, there exists an isometry mapping $q_i$ to $p_i$ for all $i\in V(G)$. The notion of universal completability ...

On Sets Defining Few Ordinary Circles

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least $n^2/4 - O(n)$ ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this ...

Regular Maps of High Density

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a simple graph with ratio between vertex-degree and number of vertices strictly exceeding $\frac{1}{2}$. We conclude that all regular maps of this type belong to a family of maps naturally ...

On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman

In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii $r_1$, $\ldots$, $r_n$ in the plane, it is always possible to cover them by a disk of radius $R = \sum r_i$, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the ...

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fréchet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original $O(n^2 \log n)$ algorithm by Alt and Godau for computing the Fréchet ...

Hyperbolicity vs. Amenability for Planar Graphs

The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.

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 ${\text {b}}(S)$ of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio ${\text {c}}(S)$ of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We ...

Straight Skeletons and Mitered Offsets of Nonconvex Polytopes

We give a concise definition of mitered offset surfaces for nonconvex polytopes in ${\mathbbm {R}}^3$, along with a proof of existence and a discussion of basic properties. These results imply the existence of 3D straight skeletons for general nonconvex polytopes. The geometric, topological, and algorithmic features of such skeletons are investigated, including a classification ...

Computing the Fréchet Distance with a Retractable Leash

All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fréchet ...

On Computability and Triviality of Well Groups

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map $f:K\rightarrow {\mathbb {R}}^n$ on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within $L_\infty$ distance r from f for a given $r>0$. The main drawback of the approach is ...

Apollonian Ball Packings and Stacked Polytopes

We investigate in this paper the relation between Apollonian d-ball packings and stacked $(d+1)$-polytopes for dimension $d\ge 3$. For $d=3$, the relation is fully described: we prove that the 1-skeleton of a stacked 4-polytope is the tangency graph of an Apollonian 3-ball packing if and only if there is no six 4-cliques sharing a 3-clique. For higher dimension, we have some ...

Metric Properties of Semialgebraic Mappings

We give various quantitative versions of Łojasiewicz inequalities for semialgebraic sets and mappings, both in the local and global case.

Fractional and j-Fold Coloring of the Plane

We present results referring to the Hadwiger–Nelson problem which asks for the minimum number of colors needed to color the plane with no two points at distance 1 having the same color. Exoo considered a more general problem concerning graphs $G_{[a,b]}$ with $\mathbb {R}^2$ as the vertex set and two vertices adjacent if their distance is in the interval [a, b]. Exoo ...

On the Densest Packing of Polycylinders in Any Dimension

Using transversality and a dimension reduction argument, a result of Bezdek and Kuperberg is applied to polycylinders, showing that the optimal packing density of $\mathbb {D}^2\times \mathbb {R}^n$ equals $\pi /\sqrt{12}$ for all natural numbers n.

Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2k with odd k, unifying and extending previous work on this subject. ...

Coloring Triangle-Free Rectangle Overlap Graphs with $O(\log \log n)$ Colors

Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $\Theta (\log \log n)$. Essentially the same construction produces $\Theta (\log \log n)$-chromatic triangle-free intersection graphs of a variety of other geometric shapes—those belonging to any class of compact arc-connected sets in ...

Triangle-Free Geometric Intersection Graphs with No Large Independent Sets

It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices.

Coloring $d$ -Embeddable $k$ -Uniform Hypergraphs

This paper extends the scenario of the Four Color Theorem in the following way. Let $\fancyscript{H}_{d,k}$ be the set of all $k$-uniform hypergraphs that can be (linearly) embedded into $\mathbb {R}^d$. We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in $\fancyscript{H}_{d,k}$. For example, we can prove that for $d\ge 3$ there ...