Centroids, centers, medians: What is the difference?
type: Item
instance of: scholarly article
title: Centroids, centers, medians: What is the difference?
zbMATH Open document ID: 0911.52006
DOI: 10.1023/A:1004934900506
author: Marco Baronti
author: Emanuele Casini
author: Pier Luigi Papini
published in: Geometriae Dedicata
publication date: 1999-04-28T00:00:00Z
review text: For a finite subset \(F\) of a real normed space, the authors give estimates for the relative distances of three types of ``central'' points of \(F\): the centroid (or center of gravity) \(g\), center \(c\) (a point where a minimum \(r\) for the maximum distance from the points in \(F\) is attained), and median \(m\) (a point where a minimum \(\mu\) for the average distance from the points in \(F\) is attained). Most notably, they show that for a finite-dimensional strictly convex space, the estimate \(\| m-c\| \leq 2 r\) is tight; for a uniformly convex space with modulus of convexity \(\delta_X(\varepsilon)\), the general estimate can be improved to \(\| m-c\| \leq 2 r (1-\delta_X(1))\). For the more intuitive case of a Hilbert space, the much smaller estimate \(\| m-c\| \leq r\) holds. A number of related open problems, modifications, and generalizations are discussed.
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 46B99
Mathematics Subject Classification ID: 90B85
zbMATH DE Number: 1098481
zbMATH Keywords: center
zbMATH Keywords: centroid
zbMATH Keywords: median
zbMATH Keywords: Weber problem
zbMATH Keywords: location
zbMATH Keywords: Banach spaces
reviewed by: Sándor P. Fekete
MaRDI profile type: Publication
full work available at URL: a:1004934900506
OpenAlex ID: W27982388
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description: scientific article; zbMATH DE number 1098481