<p>The minimum value of \(|z+1+i|+|z-1-i|+|z-1+i|+|z+1-i|\) for \(z\in\mathbb{C}\) is:</p>
Step-by-Step Solution
Key Concept: The four points are the vertices of a square with diagonal 2\sqrt{2.} The minimum sum of distances from any point to all four vertices of a square is 4\sqrt{2} (achieved at the centre).
<p>The four fixed points are \(\pm 1\pm i\) — vertices of a square centred at 0 with side \(\sqrt{2}\cdot 2/... \). At \(z=0\): sum \(= 4\sqrt{2}\). By convexity this is the minimum. ✓</p>
Correct Answer: A