Given a finite configuration of points A in ℝ k endowed with the Manhattan distance, we prove that the ratio of the sum of the distances from a centroid of A over the sum of the distances from the Steiner center of A is bounded by 1 + (k - 1) k; further, this bound can be attained. This fact extends to an arbitrary finite dimension k ≥ 2 a result proved by Fekete and Meijer for k ∈ {2, 3}.
An extension to R^k of a result by Fekete and Meijer
URSINO, PIETRO
2012-01-01
Abstract
Given a finite configuration of points A in ℝ k endowed with the Manhattan distance, we prove that the ratio of the sum of the distances from a centroid of A over the sum of the distances from the Steiner center of A is bounded by 1 + (k - 1) k; further, this bound can be attained. This fact extends to an arbitrary finite dimension k ≥ 2 a result proved by Fekete and Meijer for k ∈ {2, 3}.
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