Journal of Convex Analysis 24 (2017), No. 1, 199--212
Copyright Heldermann Verlag 2017
Defining a Unique Median via Minimizing Families of Norms
Jeffrey Tsang
Dept. of Mathematics and Statistics, University of Guelph, 50 Stone Road East, Guelph N1G 2W1, Canada
jtsang02@uoguelph.ca
Rajesh Pereira
Dept. of Mathematics and Statistics, University of Guelph, 50 Stone Road East, Guelph N1G 2W1, Canada
pereirar@uoguelph.ca
It is well-known that the median of an even number of datapoints is not unique; by any of many equivalent definitions, any point in the interval between the innermost points qualify. Recalling that the mean can be defined by a least squares approximation to the dataset, the median via least absolute differences, we consider minimizing the Lp norm from the dataset to the diagonal, and compute its limit as p approaches 1 from the right side --- the result is not the midpoint as typically used. We also construct a different family of strictly convex norms converging to L1 exhibiting a different limit-median.
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