Estimate constant under absolute loss

Imagine you have observed target values $y_1, y_2, \dots, y_n$, and you want to fit the most basic model that gives the same constant prediction for every observation:

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

Define the objective function with absolute error:

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

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

Overview:

This question tests understanding of L1 loss and robust point estimation, covering statistical concepts such as absolute deviation, order statistics, and properties of estimators under non-quadratic loss, and it belongs to the Statistics & Math domain for data scientist roles.