We study intersections of projective convex sets in the sense of Steinitz. In a projective space, an intersection of a nonempty family of convex sets splits into multiple connected components each of which is a convex set. Hence, such an intersection is called a multi-convex set. We derive a duality for saturated multi-convex sets: there exists an order anti-isomorphism between nonempty saturated multi-convex sets in a real projective space and those in the dual projective space. In discrete geometry and computational geometry, these results allow to transform a given problem into a dual problem which sometimes is easier to solve. This will be pursued in a later paper.