In this talk, I will discuss when two probability measures on the unit sphere can be transported to one another using the Gauss map of a convex body. Here, a convex body is a compact convex subset of the Euclidean n-space with non-empty interior. Notice that the boundary of a convex body might not be smooth---in general, it can even contain a fractal structure. This problem can be viewed as the problem of reconstructing a convex body using partial data regarding its Gauss map. When smoothness is assumed, it reduces to a Monge-Ampere type equation on the sphere. However, in this talk, we will work with generic convex bodies and talk about how variational argument can work in this setting. \\ \\ This is joint work with K\'aroly B"{o}r"{o}czky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang.