We have the result that all closed geodesics on $S^n$ must be contained with the intersection of $S^n$ and a plane. Hence all length minimising closed geodesics are single points.
If we equip $\mathbb{R}P^n : = \frac{S^n}{x \sim -x}$ with the metric that is the restriction of the metric on the sphere, can we say anything similar about the closed geodesics on $\mathbb{R}P^n$?