We say that a curve $ \alpha: I\rightarrow \mathbb{S}^2 $ is "sectional" if $ \alpha $ is a simple closed such that $\alpha$ divides $\mathbb{S}^2$ into two regions of equal area. I'm interested in showing that $$\inf \{L(\alpha):\alpha \ \ sectional\}\geq 2\pi$$ where $ L(\alpha) $ is the length of $\alpha$.
I appreciate any suggestions