How would I go about minimizing the expression

$\left(|z_1| + |z_2|\right) \times \left(|z_1 + z_2| + |z_1 - z_2|\right)$

subject to the constraint

$|z_1|^2 + |z_2|^2 = 1$

given that $z_1$ and $z_2$ can be complex numbers?

I thought of trying Lagrange multipliers, but it doesn't seem possible because there are an infinite number of solutions (and solving a 5-equation, 5-variable system is a bit painful).

Any hints on how I could do this?

While they are both countable, there is a more applicable mechanism for measuring how many natural numbers have a certain property. If $A \subset \mathbb{N}$, let $$A_n = \text{number of elements in A less or equal to n}$$ then we can define density of $A$ as $$\operatorname{density}(A) = \lim_{n\to\infty}  \frac{A_n}{n}$$Then the odd numbers have density $\frac{1}{2}$.