A Curious Result on Breaking Ties among Sample Medians

It is well known that any sample median value (not necessarily unique) minimizes the empirical $L^1$ loss. Interestingly, we show that the minimizer of the $L^{1+epsilon}$ loss exhibits a singular phenomenon that provides a unique definition for the sample median as $epsilon rightarrow 0$. This definition is the unique point among all candidate median values that balances the $logarithmic$ moment of the empirical distribution. The result generalizes directly to breaking ties among sample quantiles.