How Can I Solve The Following problem with $ {L}_{1} $ Approximation or Method of Least Squares'?

Question

Find the point $(x,y)$ that minimizes

$$f(x,y) = \sum_{i=1}^{n} \sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}} $$

where $(x_i, y_i)$ are chosen symmetrically with respect to the origin. Take $n$ as an even number greater than or equal to $20$.

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I think I can solve it through 'least squares method' but I want to know how to do it with $l_{1}$ approximation.

Answer 1

The function $\phi((x,y)) = f(x,y)$ is convex and $\phi(-(x,y)) = \phi((x,y))$, so $\phi((0,0)) \le {1 \over 2} (\phi(-(x,y)) + \phi((x,y))) = \phi((x,y))$.

In particular, $(0,0)$ is a solution.

Answer 2

You selection creates a distribution which is isotropic. Hence its mean and median (Which minimizes the $ {L}_{2} $ and $ {L}_{1} $ norms respectively) are given by $ \left( 0, 0 \right) $.