Sharp affine stability estimates for Hammer's problem

We prove that the area distance between two convex bodies K and K′ with the same parallel X-rays in a set of n mutually non parallel directions is bounded from above by the area of their intersection, times a constant depending only on n. Equality holds if and only if K is a regular n-gon, and K′ is K rotated by π/n about its center, up to affine transformations. This and similar sharp affine invariant inequalities lead to stability estimates for Hammer's problem if the n directions are known up to an error, or in case X-rays emanating from n collinear points are considered. For n=4, the order of these estimates is compared with the cross ratio of given directions and given points, respectively.