# Weighted orthogonal linear *L*_{∞}-approximation and applications

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## Abstract

Let *S*={*s*_{1}, *s*_{2},...,*s*_{ n }} be a set of sites in *E*^{ d }, where every site *s*_{ i } has a positive real weight *ω*_{ i }. This paper gives an algorithm to find a *weighted orthogonal L*_{∞}-*approximation hyperplane* for *S*. The algorithm is shown to require *O*(*n*log*n*) time and *O*(*n*) space for *d*=2, and *O*(*n*^{[d/2+1]}) time and *O*(*n*^{[(d+1)/2]}) space for *d*>2. The *L*_{∞}-approximation algorithm will be adapted to solve the problem of finding the width of a set of *n* points in *E*^{ d }, and the problem of finding a stabbing hyperplane for a set of *n* hyperspheres in *E*^{ d } with varying radii. The time and space complexities of the width and stabbing algorithms are seen to be the same as those of the *L*_{∞}-approximation algorithm.

## Keywords

Convex Hull Time Algorithm Feasibility Region Convex Polytope Supporting Hyperplane## Preview

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