Extremal problems for the central projection

We consider a projection from the center of the unit sphere to a tangent space of it, the central projection, and study two area minimizing problems of the image of a closed subset in the sphere. One of the problems is the uniqueness of the tangent plane that minimizes the area for an arbitrary fixed subset. The other is the shape of the subset that minimizes the minimum value of the area. We also study the similar problems for the hyperbolic space.