Systolic inequalities on the sphere from symplectic embeddings

We use properties of symplectic capacities that were recently defined by Hutchings to obtain upper bounds on the minimal action of Reeb orbits on fiberwise star-shaped hypersurfaces $\mathrm{\Sigma }\subset T^{\ast }{S}^2$. In addition, we introduce the notion of a fiberwise $\beta$-balanced hypersurface $\mathrm{\Sigma }\subset T^{\ast }{S}^2$ and establish upper bounds for the systole in terms of $\beta$ and geometric data, in the case of Riemannian metrics on $S^2$ satisfying this property. Finally, under the assumption of antipodal symmetry, we provide a non-sharp estimate of how fiberwise balanced a $\delta$-pinched metric is.