相关论文: Computational Geometry Column 44
We prove that any convex body in the plane can be partitioned into $m$ convex parts of equal areas and perimeters for any integer $m\ge 2$; this result was previously known for prime powers $m=p^k$. We also discuss possible…
This paper proposed a method to judge whether the point is inside or outside of the simple convex polygon by the intersection of the vertical line. It determined the point to an area enclosed by two straight lines, then convert the problem…
Although intersection homology lacks a ring structure, certain expressions (called uniform) in the intersection homology of an irreducible projective variety $X$ always give the same value, when computed via the decomposition theorem on any…
It is well established that a general pair of twisted cubic curves in complex projective space has ten common secant lines. As an initial investigation, we show that the monodromy group of the ten common secant lines over the complex…
A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the…
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to the sphere and the hyperbolic…
We describe an infinite family of edge-decompositions of complete graphs into two graphs, each of which triangulate the same orientable surface. Previously, such decompositions had only been known for only a few complete graphs. These…
Given a homogenous Poisson point process in the plane, we prove that it is possible to partition the plane into bounded connected cells of equal volume, in a translation-invariant way, with each point of the process contained in exactly one…
We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of…
In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the non-empty subsets $A\subset \{1,..,n\}$, is it possible to construct a body $T\subset \mathbb{R}^n$ such that $x_A=|T_A|$ where $|T_A|$…
We give a review of results on the minimum convex cover and maximum hidden set problems. In addition, we give some new results. First we show that it is NP-hard to determine whether a polygon has the same convex cover number as its hidden…
Let G be the graph of a triangulated surface $\Sigma$ of genus $g\geq 2$. A cycle of G is splitting if it cuts $\Sigma$ into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding…
We collect some open problems about minimal presentations of numerical semigroups and, more generally, about defining ideals and free resolutions of their semigroup rings and associated graded rings. We emphasize both long-standing problems…
The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as…
We propose a successive generation of cutting inequalities for binary quadratic optimization problems. Multiple cutting inequalities are successively generated for the convex hull of the set of the optimal solutions $\subset \{0, 1\}^n$,…
In this study, a geometric version of an NP-hard problem ("Almost $2-SAT$" problem) is introduced which has potential applications in clustering, separation axis, binary sensor networks, shape separation, image processing, etc. Furthermore,…
Gr\"unbaum's equipartition problem asked if for any measure $\mu$ on $\mathbb{R}^d$ there are always $d$ hyperplanes which divide $\mathbb{R}^d$ into $2^d$ $\mu$-equal parts. This problem is known to have a positive answer for $d\le 3$ and…
Punctured polygons are polygons with internal holes which are also polygons. The external and internal polygons are of the same type, and they are mutually as well as self-avoiding. Based on an assumption about the limiting area…
We study the relationship between the areas of the consecutive quadrilaterals cut from a convex quadrilateral in the plane by means of a finite or infinite number of straight lines intersecting two of its opposite sides. Moreover, we obtain…
We consider the classic problem of fairly dividing a heterogeneous good ("cake") among several agents with different valuations. Classic cake-cutting procedures either allocate each agent a collection of disconnected pieces, or assume that…