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The diagonal Δ of R n is Chebeshev with respect to the p-norm for every p ∈ (1, ∞) but not for p = 1. As a result, the median is multi-valued, since the median of a data set {a1, ··· ,an} can be thought of as the number(s) μ for which the point (μ, ··· ,μ) is a point on Δ that best approximates the point (a1, ··· ,an) with respect to the  1 -norm. In this note, it is proved that if (μp, ··· ,μp) is the unique point on Δ that best approximates a fixed point (a1, ··· ,an) with respect to thep -norm for p ∈ (1, ∞) ,t hen asp decreases to 1, μp converges, and its limit is proposed to be called the median of {a1, ··· ,an}. Along the way, μp is shown to be continuous in p for all p ∈ (1, ∞) in the sense that μp converges to μq as p goes to q for every q ∈ (0, ∞) .

Towards a well-defined median