Editing Central tendency (section)

===Uniqueness===
The mean (''L''<sup>2</sup> center) and midrange (''L''<sup>∞</sup> center) are unique (when they exist), while the median (''L''<sup>1</sup> center) and mode (''L''<sup>0</sup> center) are not in general unique. This can be understood in terms of [[convex function|convexity]] of the associated functions ([[coercive function]]s).

The 2-norm and ∞-norm are [[strictly convex function|strictly convex]], and thus (by convex optimization) the minimizer is unique (if it exists), and exists for bounded distributions. Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point.

The 1-norm is not ''strictly'' convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. Correspondingly, the median (in this sense of minimizing) is not in general unique, and in fact any point between the two central points of a discrete distribution minimizes average absolute deviation.

The 0-"norm" is not convex (hence not a norm). Correspondingly, the mode is not unique – for example, in a uniform distribution ''any'' point is the mode.