Defining a Unique Median via Minimizing Families of Norms

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 $\mathcal{L}_p$ norm from the dataset to the diagonal, and compute its limit as $p\to 1^+$ — the result is not the midpoint as typically used. We also construct a different family of strictly convex norms converging to $\mathcal{L}_1$ exhibiting a different limit-median