Centroids, centers, medians: What is the difference?
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.
scientific article; zbMATH DE number 1098481