Estimate constant under absolute loss

Suppose you have observed target values $y_1, y_2, \dots, y_n$, and you want to fit the simplest possible model that predicts the same constant value for every observation:

$\hat y_i = c \quad \text{for all } i.$

Define the objective function using absolute error:

$E(c) = \sum_{i=1}^{n} |y_i - c|.$

What value of $c$ minimizes $E(c)$? Derive the result and explain whether the minimizer is always unique.