Matheron's Conjecture for the Covariogram Problem

The covariogram of a convex body K provides the volumes of the intersections of K with all its possible translates. G. Matheron asked in 1986 whether this information determines K among all convex bodies, up to certain known ambiguities. We prove that this is the case if K\subset R^2 is not C^1, or it is not strictly convex. We construct examples that show that Matheron's question has a negative answer in R^n, for any n\geq 4. We also prove that when two planar convex bodies H and K with equal covariogram are such that the intersection of their boundaries contains an open arc then H is a translation or a reflection of K