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?
Best Answer
EDIT: my first try found the maximum rather than the minimum.
Suppose wlog $|z_1| \ge |z_2|$. Fix $z_1$ and $|z_2|$ for the moment, and write $z_2 = w z_1$. Then we want to minimize $|z_1 + z_2| + |z_1 - z_2| = |z_1| (|w + 1| + |w - 1|)$ subject to $|w| = |z_2|/|z_1|$. Geometrically, $|w + 1| + |w - 1|$ is the sum of the distances from $w$ to $1$ and $-1$, which is $2$ on the line segment $[-1,1]$ and greater than $2$ otherwise, and $w$ is on a circle centred at $0$ with radius less than $1$, so the minimum value is attained at $w = \pm |z_2|/|z_1|$. Now write $|z_1| = \cos \theta$ and $|z_2| = \sin \theta$, $0 \le \theta \le \pi/4$. We have $|z_1 + z_2| + |z_1 - z_2| = 2 \cos \theta$, so the objective is to minimize $2 (\cos \theta + \sin \theta)\cos \theta$ for $0 \le \theta \le \pi/4$, and it is easy to see that the minimum value is $0$, attained at both $\theta=0$ and $\theta=\pi/4$.