Compact families of Jordan curves and convex hulls in three dimensions

A Crossing Lemma for Families of Jordan Curves with a Bounded Intersection Number

A family of closed simple (i.e., Jordan) curves is $m$-intersecting if any pair of its curves have at most $m$ points of common intersection. We say that a pair of such curves touch if they intersect at a single point of common tangency. In this work we show that any $m$-intersecting family of $n$ Jordan curves in general position in the plane contains $O\left(n^{2-\frac{1}{3m+15}}\right)$ touching pairs Furthermore, we use the string separator theorem of Fox and Pach in order to establish the following Crossing Lemma for contact graphs of Jordan curves: Let $Γ$ be an $m$-intersecting family o…

Relative Convex Hull Determination from Convex Hulls in the Plane

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use suc…

Coloring Jordan regions and curves

A Jordan region is a subset of the plane that is homeomorphic to a closed disk. Consider a family $\mathcal{F}$ of Jordan regions whose interiors are pairwise disjoint, and such that any two Jordan regions intersect in at most one point. If any point of the plane is contained in at most $k$ elements of $\mathcal{F}$ (with $k$ sufficiently large), then we show that the elements of $\mathcal{F}$ can be colored with at most $k+1$ colors so that intersecting Jordan regions are assigned distinct colors. This is best possible and answers a question raised by Reed and Shepherd in 1996. As a simple co…

实例最优模糊凸包

Instance-Optimal Imprecise Convex Hull

该研究提出了一种同时优化检索次数和运行时间的方法，用于构造不精确点集的凸包。