Let $S$ be a hypersurface enclosed inside the unit sphere in $R^n$. We may assume that every ray $\{t x: t \geq 0 \}$ intersects $S$ at most once.

Under what extra condition is ${\rm Area}(S) \leq {\rm Area}(S^{n-1})$ ?

(I am mostly interested in the 2-dimensional case.)

Thanks.