Rigidity of projective area widths and a weighted Crofton formula on $R{P}^3$

We show that a Riemannian metric on $R{P}^3$ is a surface Zoll metric if and only if its four projective area widths are equal. We also establish a weighted Crofton formula on $R{P}^3$ under surface Zoll metrics, from which we derive an inequality relating the systole, the volume, and the common value of the projective area widths.