Related papers: 'Fair' Partitions of Polygons - an Introduction

Let $\mathbb{P}_K(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $K$, a non-flat compact convex polygon in $\mathbb{R}^2$, are in convex position, that is, form the vertex set of a convex…

We show that minimizing a convex function over the integer points of a bounded convex set is polynomial in fixed dimension.

We model the societal task of redistricting political districts as a partitioning problem: Given a set of $n$ points in the plane, each belonging to one of two parties, and a parameter $k$, our goal is to compute a partition $\Pi$ of the…

Every partition of [[omega_1]^{< omega}]^2 into finitely many pieces has a cofinal homogeneous set. Furthermore, it is consistent that every directed partially ordered set satisfies the partition property if and only if it has finite…

We prove identities generating higher dimensional vector partitions. We derive theorems for integer lattice points in the 2D first quadrant, then generalize the approach to find 3D and $n$-space lattice point vector region extensions. We…

We confirm two conjectures of Lassak on the area of reduced spherical polygons. The area of every reduced spherical non-regular $n$-gon is less than that of the regular spherical $n$-gon of the same thickness. Moreover, the area of every…

We will first solve the following problem analytically: given a piece of wire of specified length, we will find where the wire should be cut and bent to form two regular polygons not necessarily having the same number of sides, so that the…

A question of interest in Linear Algebra is whether all n x n complex matrices can be unitarily tridiagonalised. The answer for all n not equal to 4 (affirmative or negative) has been known for a while, whereas the case n=4 seems to have…

Jigsaw puzzle solving requires the rearrangement of unordered pieces into their original pose in order to reconstruct a coherent whole, often an image, and is known to be an intractable problem. While the possible impact of automatic puzzle…

We show that every heptagon is a section of a $3$-polytope with $6$ vertices. This implies that every $n$-gon with $n\geq 7$ can be obtained as a section of a $(2+\lfloor\frac{n}{7}\rfloor)$-dimensional polytope with at most…

Let $B$ be an infinite subset of $\mathbf{N}$. When we consider partitions of natural numbers into elements of $B$, a partition number without a restriction of the number of equal parts can be expressed by partition numbers with a…

In this paper we give a formula for the probability that $n$ random points chosen under the uniform distribution in a disk are in convex position. While close, the formula is recursive and is totally explicit only for the first values of…

Suppose that a polygon $P$ is given as an array containing the vertices in counterclockwise order. We analyze how many vertices (including the index of each of these vertices) we need to know before we can bound $P$, i.e., report a bounded…

We study a game where two players take turns selecting points of a convex geometry until the convex closure of the jointly selected points contains all the points of a given winning set. The winner of the game is the last player able to…

We present filling as a new type of spatial subdivision problem that is related to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most…

An $N$-dimensional parallelepiped will be called a bar if and only if there are no more than $k$ different numbers among the lengths of its sides (the definition of bar depends on $k$). We prove that a parallelepiped can be dissected into…

A vector space partition $\mathcal{P}$ in $\mathbb{F}_q^v$ is a set of subspaces such that every $1$-dimensional subspace of $\mathbb{F}_q^v$ is contained in exactly one element of $\mathcal{P}$. Replacing "every point" by "every…

There are many space subdivision and space partitioning techniques used in many algorithms to speed up computations. They mostly rely on orthogonal space subdivision, resp. using hierarchical data structures, e.g. BSP trees, quadtrees,…

We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other…

A polyiamond is a polygon composed of unit equilateral triangles, and a generalized deltahedron is a convex polyhedron whose every face is a convex polyiamond. We study a variant where one face may be an exception. For a convex polygon P,…