Locating an obnoxious plane

J.M. Díaz-Báñez, M.A. López, J.A. Sellarès
2006 European Journal of Operational Research  
Let S be a set of n points in three-dimensional Euclidean space. We consider the problem of positioning a plane π intersecting the convex hull of S such that min{d(π, p); p ∈ S} is maximized. In a geometric setting, the problem asks for the widest empty slab through n points in space, where a slab is the open region of IR 3 that is bounded by two parallel planes that intersect the convex hull of S. We give a characterization of the planes which are locally optimal and we show that the problem
more » ... n be solved in O(n 3 ) time and O(n 2 ) space. We also consider several variants of the problem which include constraining the obnoxious plane to contain a given line or point and computing the widest empty slab for polyhedral obstacles. Finally, we show how to adapt our method for computing a largest empty annulus in the plane, improving the known time bound O(n 3 log n) [8] .
doi:10.1016/j.ejor.2005.02.048 fatcat:ldwgirejirab5p5ipqfjnnchue