Consider k points x₁, . . . , x_k in R². For a given positive number d, we define the k-ellipse with radius d as the set of points x ∈ R² such that the sum of the distances from x to the points x_i is equal to d.

1. How do k-ellipses look like when k = 1 or k = 2? For k = 2, show that you can assume x₁ = –x₂ = p, ΙΙpΙΙ₂ = 1, and describe the set in a orthonormal basis of Rⁿ such that p is the first unit vector.

2. Express the problem of computing the geometric median, which is the point that minimizes the sum of the distances to the points x_i, i = 1, . . . , k, as an SOCP in standard form.

3. Write a code with input X = (x₁, . . . , x_k) ∈ R^2,k and d > 0 that plots the corresponding k-ellipse.