Algorithms for ball hulls and ball intersections in strictly convex normed planes

TL;DR

This paper extends efficient algorithms for computing ball hulls and intersections from Euclidean to strictly convex normed planes, also addressing the 2-center problem with constrained circles.

Contribution

It introduces $O(n \,\log\, n)$ algorithms for ball hulls and intersections and $O(n^2)$ solutions for the 2-center problem in strictly convex normed planes, generalizing prior Euclidean results.

Findings

Ball hulls and intersections can be computed in $O(n \log n)$ time.

The 2-center problem with constrained circles is solvable in $O(n^2)$ time.

Results suggest potential extensions to more general normed planes.

Abstract

Extending results of Hershberger and Suri for the Euclidean plane, we show that ball hulls and ball intersections of sets of $n$ points in strictly convex normed planes can be constructed in $O(n \log n)$ time. In addition, we confirm that, like in the Euclidean subcase, the $2$-center problem with constrained circles can be solved also for strictly convex normed planes in $O(n^2)$ time. Some ideas for extending these results to more general types of normed planes are also presented.