The 2-center problem and ball operators in strictly convex normed planes

TL;DR

This paper extends the 2-center problem from Euclidean to strictly convex normed planes, providing algorithms and theoretical results for ball hulls and intersections in these spaces.

Contribution

It generalizes the 2-center problem algorithm to strictly convex normed planes and proves new theorems on ball hulls and intersections in these spaces.

Findings

Decision problem solvable in O(n^2 log n) time

Generalization of Euclidean 2-center algorithm to normed planes

New theorems on ball hulls and intersections in strictly convex normed planes

Abstract

We investigate the 2-center problem for arbitrary strictly convex, centrally symmetric curves instead of usual circles. In other words, we extend the 2-center problem (from the Euclidean plane) to strictly convex normed planes, since any strictly convex, centrally symmetric curve can be interpreted as (unit) circle of such a normed plane. Thus we generalize the respective algorithmical approach given by J. Hershberger for the Euclidean plane. We show that the corresponding decision problem can be solved in $O(n^2\log\, n)$ time. In addition, we prove various theorems on the notions of ball hull and ball intersection of finite sets in strictly convex normed planes, which are fundamental for the 2-center problem, but also interesting for themselves.