  Given samples $x_1,\cdots,x_n$, it is well known that any sample median value
(not necessarily unique) minimizes the absolute loss $\sum_{i=1}^n |q-x_i|$.
Interestingly, we show that the minimizer of the loss
$\sum_{i=1}^n|q-x_i|^{1+\epsilon}$ exhibits a singular perturbation behaviour
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 when
the quantile regression loss is modified in the same way.
