Intersecting convex sets by rays

Abstract

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Coverage with k-Transmitters in the Presence of Obstacles

“…In the case of orthogonal polygons, one n 3 -transmitter is sufficient to cover the entire polygon. The problem of covering the plane with a single k-transmitter has been also considered in [12], where it is proved that there exist collections of n pairwise disjoint equal-length segments in the Euclidean plane such that, from any point, there is a ray that meets at least 2n/3 of them (roughly). While the focus in [10,2,12] is on finding a small number of high power transmitters, our focus in this paper is primarily on lower power transmitters.…”

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…The problem of showing the existence of a point with large ray-shooting depth is open in higher dimensions. Other notions of ray-shooting depth for convex sets, instead of points sets, were studied in [173]. Algorithms.…”

A time–space trade-off for computing the k-visibility region of a point in a polygon

“…The notion of k-visibility has previously been considered in the context of art-gallerystyle questions [5,13,16,22] and in the definition of certain geometric graphs [11,15,18]. While the 0-visibility region is always connected, the k-visibility region may have several components.…”