On the 2-Center Problem Under Convex Polyhedral Distance Function

Abstract

Metrics or distance functions generalizing the Euclidean metric (and even $L_p$-metrics) have been used in Computational Geometry long ago; for example, convex distance functions appeared in the first Symposium on Computational geometry [2]. Very recently Das et al. [5] studied the 1-center problem under a convex polyhedral distance function where the unit ball of the distance function is a convex polytope. In this paper we develop algorithms for the 2-center problem under a convex polyhedral distance function in $\mathbb {R}^d,d=2,3$. We show that the 2-center can be computed in $O(n\log m)$ time for the plane and in $O(nm\log n)$ time for $d=3$. We show that the problem of for computing the 2-center in $\mathbb {R}^3$ has an $\varOmega (n\log n)$ lower bound.